Target

Fundamental Small Question 2 (Answer)

2010-03-22T16:32:00.002+05:30

Answer of Fundamental Small Question 2

1. How is character constant represented?

Ans:- A Character constants is written within single quotes. For example, ‘K’, ‘\’, and ‘5’.

2. How is string constant represented?

Ans:- A string constants is a sequence of zero or more characters enclosed within double quotes. For example, “X”, “656”, “ ”, and “RAJ”.

3. List the printable escape sequences.

Ans:- These are printable escape sequence : \\, \”, \’, and \?.

4. List the non graphic escape sequences.

Ans:- These are the non-graphic escape sequence :

\0 :- null character

\a :- alert

\b :- backspace

\t :- horizontal tab

\n :- new line

\v :- vertical tab

\f :- form feed

\r :- carriage return

5. What is the symbolic constant?

Ans:- If a constants is given a name, it becomes a symbolic constant or manifest. For example,

#define MAX 20

Defines the symbolic constant MAX to represent the constant value 20.

6. How is variable declared?

Ans:- Format (syntax):-

Data_type var1, var2, var3,……..,varn;

Ex.:- int a, b; float x, y;

7. What is initialization of a variable?

Ans:- If a variable is assigned a value in the declaration itself, it is known as initialization. For example,

Int x = 10, y = 5;

8. How are comments included in a C program?

Ans:- Single line or multiline comments are included between /* and */. Nested comments are not allowed in ANSI C. For example,

/* This is a function */

9. List white space characters in C.

Ans:- Blank space, new line, horizontal tab, vertical tab, carriage return and form feed are white space characters in C.

10. What is statement terminator? What is separator?

Ans:- Each statement is ended with a semicolon, so a semicolon is a statement terminator. Whereas, a white space character or a comment is a separator used to separate tokens.

11. List the token in C.

Ans:- Identifiers, keywords, constants, string constants, operators and separators are C tokens.

12. Identify the tokens in the expression if(mark >= 50).

Ans:- Here:

The keyword if is a token, the character ( is the next token, the variable mark is the subsequent token, the operator >= is the next token, the constant 50 is the next token, and the character ) is the next token.

13. What are lvalue and rvalue?

Ans:- Each variable has got two values, which are left value and right value whose short names are lvalue and rvalue. The lvalue denotes an object that is the address of the data object. The rvalue is the value residing in that address. An lvalue represents a memory location where a value may be stored; it denotes an object that refers to a storage location of the data object. The term lvalue is used, based on the principle of its appearance, on the left side of an assignment statement. Consider the following statements:

Int x, y;

X = 5;

Y = X;

In the first assignment statement, X is an lvalue. Hence, X is treated as a name for a particular memory location and the value 5 is stored in the memory location. In the second assignment statement, X is not an lvalue and hence the value stored in the memory location is referred to, by X. Thus, in some statements an operand must be an lvalue. If an lvalue appears in any other context, it is replaced by the value stored in that memory location.

14. What is the value of sizeof(char)?

Ans:- The value returned by sizeof(char) is always 1 since char uses only one byte in any machine. For other data types, the number of bytes used to represent a data type depends on the implementation of a compiler.

15. What is the current version of C?

Ans:- C99 is the current version of C. The character set is changed by supporting Unicode characters in this version. It includes additional data types, such as bool, complex, variable-length arrays, and variable structure members and some more new syntax also. Since there is a substantial change, most compilers have not yet implemented C99. The specification of new C99 may be obtained from ISO web site.

Fundamental Small Question 2

2010-03-19T03:25:00.000+05:30

Some Small Question 2

1. How is character constant represented?

2. How is string constant represented?

3. List the printable escape sequences.

4. List the non graphic escape sequences.

5. What is the symbolic constant?

6. How is variable declared?

7. What is initialization of a variable?

8. How are comments included in a C program?

9. List white space characters in C.

10. What is statement terminator? What is separator?

11. List the token in C.

12. Identify the tokens in the expression if(mark >= 50).

13. What are lvalue and rvalue?

14. What is the value of sizeof(char)?

15. What is the current version of C?

Fundamental Small Question 1 (Answer)

2010-03-17T04:38:00.008+05:30

Answer of Fundamental Small Question 1

1. Who developed C language?

Ans:- Dennis Ritchie at AT & T Bell Laboratory, Murray Hill, New Jersey Developed C in 1972.

2. Why is C named of this language?

Ans:- The “C” name of this language is represent it as the successor of B language.

3. Who developed B language?

Ans:- Ken Thompson in 1970 developed B for the first UNIX system on the DEC PDP-7 computer.

4. Give the silent features of C.

Ans:- C is general purpose, free format and structured programming language. It has a rich set of operators, and uses more control structures. Memory addresses are directly accessed by using pointers. It is quite suitable for system programming. It is flexible and powerful language. It is fast running, machine independent and efficient language.

5. What are the limitations of C?

Ans:- There is no uniformity in associativity and no direct I/O facility.

6. Give the name any five C compiler?

Ans:- The name of five ‘C’ compilers:-

(i) Turbo C

(ii) Microsoft C

(iii) Quick C

(iv) Lattice C

(v) Power C

7. Give the character set of C.

Ans:- Alphabets : A to Z and a to z

Digits : 0 to 9

Special Characters : + - * / % = < > _ blank : ; , . ‘ “ ? ! # \ ( ) [ ] { } & | ^ ~

8. List out the name of C language reserved words.

Ans:- ANSI C keywords are listed below:

auto, break, case, char, const, continue, default, do, double, else, enum, extern, float, for, goto, If, int, long, register, return, short, signed, sizeof, static, struct, switch, typedef, union, unsigned, void, volatile, while.

9. What is the purpose of a variable and a constant in a programming language?

Ans:- The data are represented using variable or constants in a programming language.

10. Give the name of scalar data types in C.

Ans:- Scalar data types in C are int, char, float and double.

11. Give the name of derived data types in C.

Ans:- Derived data types in C are array, function, pointers, structure and unions.

12. What are qualifiers in C?

Ans:- Qualifiers or modifiers are identifiers that may precede the scalar data types (except float) to specify the number of bits used for representing the respective type of data in memory. The qualifiers in C are short, long, signed and unsigned.

13. Give the name of type specifiers in C.

Ans:- Name of the type specifiers which is used in C : char, int, float, double, short, long, signed and unsigned.

14. Give the name of type of constant in C.

Ans:- The name of type of constants which is used in C : Integer constant, single and double precision constants, character constants, string constants and symbolic constants.

15. How are octal and hexa decimal constants are represented?

Ans:- Octal constants are preceded by zero and they are formed by using digits 0 to 7.

For example, 056 and 063.

Hexa constants are preceded by 0x or 0X. They are formed by using digits 0 to 9 and characters A to F or a to f.

For example, 0x532 and 0xa1c3.

Fundamental Small Question 1

2010-03-08T16:35:00.000+05:30

Some Small Question 1

1. Who developed C language?

2. Why is C named of this language?

3. Who developed B language?

4. Give the silent features of C.

5. What are the limitations of C?

6. Give the name any five C compiler?

7. Give the character set of C.

8. List out the name of C language reserved words.

9. What is the purpose of a variable and a constant in a programming language?

10. Give the name of scalar data types in C.

11. Give the name of derived data types in C.

12. What are qualifiers in C?

13. Give the name of type specifiers in C.

14. Give the name of type of constant in C.

15. How are octal and hexa decimal constants are represented?

Number System – 9

2010-03-08T16:29:00.000+05:30

Number System – Exercise – 1 (Answer)

1. Divisor = 41, dividend = 15625, quotient = 381 and remainder = 4.

2. Divisor = (dividend – remainder )/quotient = (397246 – 211)/865 = 459.

3. Divisor = (dividend – remainder )/quotient = (x + ak) – x/a = k.

4. On dividing 87375 by 698, the remainder is 125, the least number to be subtracted from the dividend in the remainder

So, the least number to be subtracted = 125.

5. On dividing 49123 by 263, the remainder is 205, the least number to be added to the dividend = divisor – remainder = 263 – 205 = 58.

So, the least number to be added = 58.

6. The greatest 3 digit number = 999, on dividing 999 by 35, remainder = 19.

So, the required number = dividend – remainder = 999 – 19 = 980.

7. The least number of 3-digits = 100. On dividing 100 by 14, remainder = 2

So, the required number = dividend + (divisor – remainder) = 100 + (14 – 2) = 112.

8. Here, the divisor 602 is multiple of the divisor 14, and greater divisor remainder is greatest then smaller divisor.

So, required number = 34/14 = 8.

9. Here, the divisor 357 is the multiple of the divisor 17, but the greater divisor remainder is smaller then smaller divisor,

So, the remainder is same which is 5.

10. The binary equivalent of these number is:-

(i) (30)¹⁰ = (11110)²

(ii) (27)¹⁰ = (11011)²

(iii) (41)¹⁰ = (101001)²

Number System 8

2010-03-07T15:09:00.001+05:30

Number System – Exercise – 1

1. On dividing 15624 by 41, what is the quotient and the remainder?

2. On dividing 397246 by a certain number, the quotient is 865 and the remainder is 211. Find the divisor.

3. What is the number which on dividing (x + ak) gives ‘a’ as the quotient and x as the remainder?

4. Find the least number, that must be subtracted from 87375, to get a number exactly divisible by 698.

5. What least number must added to 49123 to get a number exactly divisible by 263.

6. Find the greatest number of 3 digits, which is exactly divisible by 35.

7. Find the least number of 3 digits, which is exactly divisible by 14.

8. A number when divided by 602 leaves a remainder 36. What remainder would be obtained by dividing the same number by 14?

9. A number, when divided by 357, leaves a remainder 5. What remainder would be obtained by dividing the same number by 17?

10. Find the binary equivalent of

(i) 30

(ii) 27

(iii) 41

Fundamental

2010-03-07T15:02:00.002+05:30

Fundamental of C language

* C was developed by Dennis Ritchie in 1972.

* C has its own Character set and Reserved words.

* It has four basic data types int, char, float and double, and derived data type.

* Qualifiers such as short, long, signed and unsigned may precede basic data types specific memory representation of a data type.

* C supports numeric constants, character constants and string constants.

* Symbolic constants are used to give names for constants.

* An escape sequence is a special character starting with a backslash is made to escape; it can be represented in a character constant or in a string constant.

* Comments can also be include in a program to improve the readability of the program.

* Each individual unit in a C program is called token.

* A separator is used to separate tokens, whereas, a semicolon is used as a statement terminator.

Identifier

An identifier is a symbolic name used to refer to an entity such as a data type, constant, variable, function, array etc., in a program. It is a sequence of characters string with an alphabet or underscore that may be followed by alphanumeric characters and underscores.

Variable and Constants

A variable is an identifier used for storing data in a program. The value stored in a variable may be changed during the execution of a program. A constant is a fixed value directly used in any program and it is unchanged during the execution of the program.

Scalar Data Type

A scalar data type is used for representing a single value only. For example, int, char, float and double.

Derived Data Type

Derived data types are derived from the scalar data types with additional relationships between the various elements of that scalar data type. They are also known as structured data types. Derived data type may be used for representing a single value or multiple values.

Escape Sequence

An escape sequence is used to escape from the normal meaning of a character in a C program. It is used for editing non graphic characters in a program. It is a character representation that may appear in a character constant or in a string constant. For example, \n \t \a \” \\.

Token

A token is an individual entity of a program. A compiler identifies and splits a program into a number of tokens. It always constitutes the largest possible token.

ANSI C Standard

American National Standards Institute (ANSI) established a committee X3J11 in 1983 to standardize C language. The committee’s work was finally ratified as ANSI X3.159-1989 on 14^th December 1989 and published in 1990. The standard has been adopted as an international standard ISO/IEC 9899:1990 and this replaces the earlier X3.159. Now a new Standard is developed which is nicknamed C9X. It is generally known as C99 because ISO standard was formally adopted in the year 1999. The previous standard ratified in 1989 is called C89.

BASIC

2010-03-06T15:38:00.002+05:30

Some basic concept question and Answer about C Language

1. What is a Programming Paradigm?

Ans.:- Programming paradigm means the principle that is used for organizing programs. There are two major Programming Paradigms, namely Structured Programming and Object Oriented Programming (OOP). C language uses the structured Programming Paradigm, whereas C++, C#, VB.NET or Java make use of OOP. OOP has lots of advantages to offer, But even while using this organizing principle you would still need a good hold over the language elements of C and the basic programming skills.

2. Device Driver and Embedded System Applications are created in C language or not?

Ans.:- Yes, Device Drivers are programs that are used to driver different devices attached to machine. Some device driver programs don’t actually drive a device, but perform a low level task like monitoring any changes in the windows registry. All these programs are almost exclusively written in C.

Most Embedded System programs are written in C, through assembly language is also preferred by some programming for building Embedded Systems.

3. Is it true that Operating systems like Windows, Linux and Unix are written in C?

Ans.:- Major parts of popular operating systems like Windows, Unix, Linux are still written in C. This is because every today when it comes to performance (speed of execution) nothing beats C. Also, the functions exposed by the Operating System API can be easily called through any language.

Moreover, if one is to extend the operating system to work with new devices one needs to write Device Driver programs. These Programs are exclusively written in C.

4. Which compiler should I use to learn C programming?

Ans.:- If you are a beginner you would be better off using a simple compiler like Turbo C or Turbo C++. You can also use the more sophisticated and more modern compilers like Visual Studio under windows or gcc under Linux. Note that all C++ compilers are also able to compile C programs.

5. What is the different between Visual Studio Professional and Visual C++ Express Edition?

Ans.:- Visual Studio Professional is for professional software development. As against this, the Express edition compiler has been made available by Microsoft for student and learners community completely free of cost.

Number System 7

2010-03-05T17:15:00.002+05:30

Binary Number System

This number system has base 2 and use only 0 and 1; whereas the conventional decimal system having a base 10, uses 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

The positional weight of a binary number:-

Positions from

right to left

8^th

7^th

6^th

5^th

4^th

3^rd

2^nd

1^st

Positional weight of a Binary number

2⁷

128

2⁶

2⁵

2⁴

2³

2²

2¹

2⁰

By using this table:-

The decimal equivalents of any binary number can be found out.

(10101)₂ = 1 x 2⁴ + 0 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 2⁰

= 16 + 0 + 4 + 0 + 1 = (21)₁₀

Alternatively,

1	0	1	0	1
16	8	4	2	1

Hence, (10101)₂ = 16 + 4 + 1= (21)₁₀

Conversion of a Decimal Number to Binary Number

To convert a decimal number to a binary number, the following steps are to be considered.

Step 1:- Divide the decimal number by 2.

Step 2:- Go on dividing the quotients (obtained at each stage) by 2 till the quotient is 0.

Step 3:- Write down the remainders on the right side after each of the above divisions.

Step 4:- Arrange the remainders ( as obtained in Step 3) in the reverse order to get the equivalent binary number.

Ex.:- convert a decimal number (25)₁₀ to the equivalent Binary number.

Soln.:- Divide the given number 25 by 2 and go on dividing the quotient by 2 till the quotient is 0.

Then we get the remainder in every step like 1, 0, 0, 1, 1 respectively.

Now, arrange the remainder in reverse order like, 11001. This is the required Binary number.

So, (25)₁₀ = (11001)₂

Calculation in the Binary System

Mathematical calculations (i.e. addition, subtraction and multiplication) in the binary system follow their own rules and are similar to those in the decimal system.

Binary Addition

It is easy to add two binary numbers. The rules for binary addition are as follows:-

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10 (put 0 to the same column and carry 1 to the next left column)

Binary Subtraction

It is easy to subtract a binary number from another binary number. The rules for binary subtraction are as follows:

0 – 0 = 0

1 – 1 = 0

1 – 0 = 1

To find 0 – 1, we write 1 in the result and also we borrow 1 from the next left column.

Binary Multiplication

Binary multiplication is similar as decimal multiplication. The four rules that are followed in multiplication of two binary numbers are following:-

0 x 0 = 0

0 x 1 = 0

1 x 0 = 0

1 x 1 = 1

Binary Division

In the division, the method that is applied is similar to that in decimal system. The two rules which are followed here are,

⁰/₁ = 0 and ¹/₁ = 1

Here also, the value of ¹/₀ is undefined.

Number System 6

2010-03-05T00:51:00.003+05:30

BASE and INDEX

If a number ‘b’ is multiplied by it self ‘n’ times, then the product is called the nth power of b, i.e.

b x b x b x b ……. Up to n times = bⁿ

Here, b is called base and n is called the index.

Last digit ( Digit at unit place) in (xyz)ⁿ

Here the given number is (xyz)ⁿ, ‘xyz’ is base and ‘n’ is the index and ‘z’ is the last digit of the base.

To find out the last digit in (xyz)ⁿ, following steps are to be followed.

Divide the index (n) by 4, then

CASE – 1

If remainder = 0, then check if ‘z’ is odd (except 5), then last digit = 1.

And if ‘z’ is even, then last digit = 6.

CASE - 2

If remainder = 1, then required last digit = last digit of base (i.e. ‘z’)

If remainder = 2, then required last digit = last digit of (z)²

If remainder = 3, then required last digit = last digit of (z)³

Note:- If z is 5, then last digit in the product = 5.

Ex.:- Find the last digit in (295073)¹³⁰

Soln.:- Dividing 130 by 4, the remainder = 2

Referring to Case - 2, the required last digit is the last digit of (z)², i.e. (3)² = 9

Number of Zeroes at the end of product

On multiplying two or more given numbers, the zeros are produced at the end of the resulting product due to the following reasons:

(a) If there is any zero at the end of the factors (or numbers being multiplied)

Ex.:- 7 x 20 =140, the zero of 20 is produce at the end of the product also.

(a) If 5 or a multiple of 5 is multiplied by any even number.

Ex.:- 45 x 12 = 540, multiple of 5 is multiplied by even number 2

Combining the above two reasons, we may say that:

(i) Resolve all the given number into their factors.

(ii) Count the number of 2s and 5s. i.e. (5)^x x (2)^y.

(iii) No. of zeroes at the end of the product = No. of the 2s or no. of 5s, whichever is less

Number System 5

2010-03-04T01:48:00.003+05:30

Test of a Prime Number

A prime number is only divisible by 1 and by the number it self. The first prime number is 2. Every prime number other then 2 is odd, but every odd number is not necessarily a prime number. To test whether any given number is a prime number or not, following steps are to be considered:

Step 1:- Find an integer (x) which approximate square root is greater then of the given number.

Step 2:- Test the divisibility of the given number by every prime number less than x.

Step 3:- If the given number is divisible by any of them in Step 2, then the given number is not a prime number.

If the given number is not divisible by any of them in Step 2, then the given number is a prime number.

Ex.:- Consider a number 203. Test if it is a prime or not.

Step 1:- The approximate square root of 203 is 14 plus, take x = 15.

Step 2:- Check the divisibility of 203 by the prime number less than 15 i.e., by 2, 3, 5, 7, 11, 13.

Step 3:- 203 is divisible by 7. Thus, it is not a prime number

SUM rules on Natural Numbers

Rule 1

Sum of the first n natural numbers (Starting from 1) = ^{n(n + 1)}/₂.

Ex.:- Sum of 1 to 74 = ^{74 x 75}/₂ = 2775

Rule 2

Sum of first n odd numbers (Starting from 1) = n²

Ex.:- Sum of first 7 odd numbers ( 1 + 3 + 5 + 7 + 9 + 11 + 13) = 7² = 49

Rule 3

Sum of first n even numbers (Starting from 1) = n(n + 1)

Ex.:- Sum of first 9 even numbers (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18) = 9(9 + 1) = 90.

Rule 4

Sum of squares of first natural numbers (Starting from 1) = ^{n(n + 1)(2n + 1)}/₆

Ex.:- Sum of squares of first 8 natural numbers

1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² = ^{8(8 + 1)(2 x 8 + 1)}/⁶

= ^{8 x 9 x 17}/₆ = 204

Rule 5

Sum of cubes of first n natural numbers = [ ^{n(n + 1)}/₂ ]²

Ex.:- Sum of cubes of first 6 natural numbers = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ = [ ^{6(6 + 1)}/₂ ]² = 441

Note:- For applying Rule 2 and Rule 3, it is required to find how many odd numbers or even numbers are there in the given series.

In the first ‘n’ natural numbers,

If n is even, then there are ⁿ/₂ odd numbers and ⁿ/₂ even number

If n is odd, then there are ^{n +}½ odd numbers and ^{n –}½ even numbers

Ex.:- From 1 to 30, as 30 is even, there are 15 odd numbers and 15 even numbers.

From 1 to 29, as 29 is odd, there are ^{29 +}½ = 15 odd numbers and ^{29 –}½ even numbers.

Number System - 4

2010-03-03T02:51:00.003+05:30

Remainder Rules

Rule 1

When the same number is dividend by the two different divisors such that one divisor is a multiple of the other divisor and also the remainder for the greater divisor is known.

If the remainder for the greater divisor = r

And the smaller divisor = d, then

That when r > d, the required remainder for the smaller divisor will be the remainder found out by dividing the ‘r’ by ‘d’.

And when r < d, then the required remainder is ‘r’ it self.

Ex.:- If a number is divided by 527, the remainder is 42. What will be the remainder if it is divided by 17?

Soln.:- Here the same number is divided by two divisors: 527 and 17.

Now, 527/17 = 31, so, 527 is a multiple of 17.

Remainder of the greater divisor = 42 and smaller divisor = 17.

So, 42/17 = 8 = required remainder for the smaller divisor.

Hence, I f the number is divided by 17, the remainder will be 8.

Rule 2

If two different number a and b, on being divided by the same divisor leave remainder ‘r1’ and ‘r2’ respectively, then their sum (a + b) if divided by same divisor will leave remainder R, given by

R = (r1 + r2) – divisor

The required remainder R = sum of remainders – divisor (when some is divided)

Note:- If ‘R’ becomes negative in the above equation, then the required remainder will be the sum of the remainder.

The required remainder = sum of remainders.

Ex.:- Two different numbers, when divided by the same divisor, leave remainders 15 and 39 respectively, and when their sum was divided by the same divisor, the remainder was 7. What is the divisor?

Soln.:- Using the rule 2

7 = (15 + 39) – divisor

Divisor = 47.

Rule 3

When two numbers, after being divided by the same divisor leave the same remainder, then the different of those two numbers must be exactly divisible by the same divisor.

Ex.:- Two numbers 147 and 225, after being divided by a 2-digit number, leave the same remainder. Find the divisor.

Soln.:- By rule 3, the difference of 225 and 147 must be perfectly divisible by the divisor.

The difference = 225 – 147 = 78

Now, 78 = 13 x 2 x 3

Thus, 1-digit divisors = 2, 3 and 2 x 3 = 6

2-digit divisors = 13, 13 x 2 = 26, 13 x 3 = 39, 13 x 2 x 3 = 78

So, the possible divisors are 13, 26, 39, 78.

Rule 4

If a given number is divided successively by the different factors of the divisor leaving remainder ‘r1’, ‘r2’, and ‘r3’ respectively, then the true remainder (i.e. remainder when the number is divided by the divisor) can be obtained by using the following formula:

True remainder = (first remainder) + (second remainder x first divisor) + (third remainder x second divisor x first divisor)

Ex.:- A number, being successively divided by 5, 7 and 11 leaves 3, 1, 2 as remainders respectively. Find the remainder if the same number is divided by 385.

Soln.:- Here, the divisor is 385, whose factors are 5, 7 and 11

By the rule 3, True remainder (i.e. remainder when divided by 385) = 3 + (1 x 5) + (2 x 5 x 7)

= 3 + 5 + 70 = 78

Rule 5

When (x + 1)ⁿ is divided by x, the remainder is always 1, where x and n are natural numbers.

Ex.:- What will be the remainder when (17)²¹ is divided by 16?

Soln.:- (17)²¹ = (16 + 1)²¹

When (16 + 1)²¹ is divided by 16, the remainder = 1.

Rule 6

When (x – 1)n is divided by x, then

The remainder = 1, when n is an even natural number

But the remainder = x-1, when n is an odd natural number

Ex.:- What will be the remainder when (29)⁷⁵ is divided by 30?

Soln.:- (29)⁷⁵ = (30 – 1)⁷⁵, here index = 75 (which is odd) so, when (30 – 1)⁷⁵ is divided by 30, the remainder will be x – 1 = 30 – 1 = 29.

Number System - 3

2010-02-28T17:08:00.001+05:30

Division and Remainder

When a given number is not exactly divisible by any number, then there is a remainder number at the end of such division.

Suppose we divide 25 by 7 as,

Then, divisor =7, dividend = 25, quotient = 3 and remainder = 4.

There is most important formula to check the divisibility:-

Dividend = (divisor x quotient) + remainder

So, if a number ‘x’ is divided by ‘k’, leaving remainder ‘r’ and giving quotient ‘q’ then the number can be found by using (i)

x = kq + r

Hence, if the number x is exactly divisible by k, then remainder = r = 0

x = kq

and so ^x/_k = q, implying that x is divisible exactly by ‘k’ and ‘q’ is an integer.

Methods to Find a Number Completely Divisible by Another

Consider a given number x. when divided by ‘d’, it gives a quotient ‘q’ and remainder ‘r’.

Now, to find a number exactly divisible by ‘d’, we can use either of the following two methods to reduce the remainder to zero. ( If a number is exactly divisible, then remainder is zero).

Method 1

By subtracting the remainder from the given number ( dividend).

The required number that is exactly divisible by ‘d’ = x – r

Hence ‘remainder’ is the least number that can be subtracted from any number to make it exactly divisible.

Method 2

By adding the (divisor – remainder) to the given number.

The required number that is exactly divisible by d = x + (divisor – remainder)

Therefore, ( divisor – remainder ) is the least number that can be added to any given number to make it exactly divisible.

Example:

Find the least number, that must be

(a) Subtract from, or

(b) Added to a given number 5029, to make it exactly divisible by 17.

Soln.:- On dividing 5029 by 17, we get remainder = 14.

(a) The least number to be subtracted to make it exactly divisible = remainder = 14.

(b) The least number to be added to make it exactly divisible = divisor – remainder = 17 – 14 = 3.

Greatest n-digit and Least n-digit number Exactly Divisible by a number

(a) To find out the greatest n –digit number exactly divisible by a divisor ‘d’, we use method 1.

The required number = greatest n-digit number – remainder.

(b) To find out the least n-digit number exactly divisible by a divisor ‘d’, we use method 2, because if we use method 1, then subtracting any number from the n-digit least number will reduce it to (n-1) digit number.

The required number = least n-digit number + (divisor – remainder)

Example:- Find the

(a) greatest 3-digit number divisible by 13.

(b) The least 3-digit number divisible by 13.

Soln.:-

(a) Greatest 3-digit “999” divided by 13, we get remainder = 11

So, the required 3-digit greates number = 999 – 11 = 988

(b) Least 3-digit “100”divided by 13, we get remainder = 9

So, the required 3-digit least number = 100 + (13 – 9) = 104.

Number System - 2

2010-02-27T01:58:00.003+05:30

After types of number system, we study about divisibility property of numbers.

There are certain tests for divisibility of numbers by any of the numbers 2, 3, 4, 5, 6, 8, 9, 10 and 11 such that by simply examining the digits in the given number, one can easily determine whether or not a given number is divisible by any of these numbers.

Some test detailed as follows:

Divisibility by 2
If the last digit is an even number or it has zero (0) at the end.
Ex.:- 74, 148, 1210 are all divisible by 2.

Divisibility by 3
If the sum of the digits of the given number is divisible by 3, then that number is fully divisible by 3.
Ex.:- The sum of the digits of number 3705 is 3 + 7 + 0 + 5=15. 15 is divisible by 3 so the number 3705 is fully divisible by 3.

Divisibility by 4
If the formed by the last two digits of the given number is divisible by 4, or if the last two digits are ‘00’, then that number is fully divisible by 4.
Ex.:-216560 is a number whose last two digits are 60, 60 is divisible by 4 so the given number 216560 is also divisible by 4.

Divisibility by 5
If the last digit of the given number is 0 or 5 then the number is fully divisible by 5.
Ex.:- 865, 1605, 5670 are all divisible by 5.

Divisibility by 6
If the given number is divisible by 2 and 3 then that number is also divisible by 6.
Ex.:- let us consider the number is 89004. It has 4 as the last digit, so it is divisible by 2.
Sum of the digits 8 + 9 + 0 + 0 + 4=21, 21 is divisible by 3.
Since, 89004 is divisible by 2 and 3 both so that number is also divisible by 6.

Divisibility by 8
If the number formed by the last three digits of the given number is divisible by 8 or if the last three digits are ‘000’, then that number is fully divisible by 8.
Ex.:- The number 56976 has 976 as the last three digits. Since 976 is divisible by 8, 56976 is also divisible by 8. The number 5463000 end with ‘000’ and so, it is divisible by 8.

Divisibility by 9
If the sum of the digits of the given number is divisible by 9, then that number is also fully divisible by 9.
Ex.:- 890676 is a number the sum of whose digits is = 8 + 9 + 0 + 6 + 7 + 6=36. Since 36 is divisible by 9 so the number 890676 is also divisible by 9.

Divisibility by 10
If the last digit of the number is zero (0), then that number is fully divisible by 10.
Ex.:- 890 has 0 at the end, so it is divisible by 10.

Divisibility by 11
If the difference of the sum of its digits in odd places and the sum of its digits in even places is either zero (0) or a multiple of 11.
Ex.:- Let us consider the number 647053.
Sum of digits at odd places = 6 + 7 + 5= 18
Sum of digits at even places = 4 + 0 + 3= 7
Difference of the sums = 18 – 7=11
Since the difference 11 is a multiple of 11, so 647053 is divisible by 11.

Some General properties of Divisibility :-

1. If a number ‘x’ is divisible by another number ‘y’, then any number divisible by ‘x’, will also be divisible by ‘y’ and by all the factors of ‘y’.
Ex.:- The number 84 is divisible by 6. Thus any number that is divisible by 84, will also be divisible by 6 and also by the factors of 6, i.e. by 2 and by 3.

2. If a number ‘x’ is divisible by two or more than two co-prime number then x is also divisible by the product of those numbers.
Ex.:- The number 2520 is divisible by 5, 4, 13 that are prime to each other (i.e. co-prime), so, 2520 will also be divisible by 20 (= 5 x 4), 52 (= 4 x 13).

3. If two number ‘x’ and ‘y’ are divisible by a number ‘p’, then their sum x + y is also divisible by ‘p’.
Ex.:- The number 225 and 375 are both divisible by 5. Thus their sum = 225 + 375 = 600 will also be divisible by 5.
Note:- It is also true for more than two numbers.

4. If two number ‘x’ and ‘y’ are divisible by a number ‘p’, then their difference x – y is also divisible by ‘p’.
Ex.:- The number 126 and 507 are both divisible by 3. Thus their difference = 507 – 126 = 381 will also be divisible by 3.

Number System - 1

2010-02-24T16:23:00.000+05:30

In aptitude we first start with the Number system.

Here, numbers are divided in some category i.e. Natural number, Whole number, Integer, rational number, irrational numbers etc.

Natural numbers:- It is start from 1 to infinity and it is non-negative numbers.
Set of natural numbers is {1, 2, 3, ……}

Whole number:- It is also non-negative number including “0”.
Natural number including “0” is called Whole Numbers.
Set of whole numbers is {0, 1, 2, 3, …….}

Integers:- All numbers are called integer it may be positive or negative.
Set of Integers is {……,-3, -2, -1, 0, 1, 2, 3,……}

Rational number:- Any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero is called Rational number.
Set of rational number is ½, 3/5, -8/5, 0, +3, -150, …….

Irrational number:- Any real numbers are called Irrational number which is not a Rational number.
Set of irrational number is π, √2,√5,√7, …….

Some numbers are like √-8,√-7,√-5,√-1,3+√-7 and so on. These are undefined numbers, called Complex numbers.

Integers- There are three types of integers :-
Positive integers:- 1, 2, 3,………
Negative integers:- -1, -2, -3, ………….
Non-negative integers:- 0, 1, 2, 3, ….

Some other numbers type:-

Even Number:- A counting number which divisible by 2 is called an even number.
Eg.:- 0, 2, 4, 6, 8, ……

Odd number:- A counting number which is not divisible by 2 is called an odd number.
Eg.:- 1, 3, 5, 7, ……….

Prime number:- A number is called a prime number if it has exactly two factors, namely itself and 1.
Eg.:- 2, 3, 5, 7, 11, 13, 17, 19, ……..

Co-primes number:- Two natural numbers a and b are said to be a co-prime if their HCF is 1.
Eg.:- (2, 3), (4, 5), (7, 9), (8, 11)… etc.

First post

2010-02-24T02:34:00.000+05:30

Today is my first post on this blog. This blog is very useful for that person who is preparing for the competitive examination for any purpose like job or admission in any institution.

This blog contents both Arithmetic aptitude and computer technical problems.

Today we start with the important of the Aptitude. Aptitude is more important to our education life. These days every ware a small aptitude test to get the admission. It is not fully based upon the pure mathematics, it is actually a technical mathematic in which we solve the problem by using small tips and tricks.

Another subject of this blog is computer technical problem, which is useful to the computer engineers. It is related to the technical test which is organized by the industry at placement or campus time.

From the next post you get the information about aptitude and computer technical.

Thank you….