Vedic Mathematics Tricks

Divisibility tests of common numbers

2011-02-26T21:18:00.001+05:30

Divisibility of different numbers has been listed below:

Divisibility by 2 : A number is divisible by 2, if its unit digits is any of 0,2,4,6,8.
In other words, when the unit's place digit is even or zero (0).
Divisibility by 3 : A number is divisible by 3 only when the sum of its digits is divisible by 3. In other words, All numbers with a digit sum of three (3), six (6) or nine (9) are divisible by 3.
Divisibility by 4 : A number is divisible by four (4), if its last two (2) digits are divisible by 4.
Divisibility by 5 : A number is divisible by 5 if its unit digit is five (5) or zero (0).
Divisibility by 6 : A number is divisible by 6 if it is divisible by both two (2) and three (3).
Divisibility by 8 : A number is divisible by eight (8) only when the number formed by its last three (3) digits is divisible by eight (8).
Divisibility by 9 : A number is divisible by nine (9) if sum of its digits is divisible by nine (9). In other words, All numbers with a digit sum is nine (9) are divisible by nine (9).
Divisibility by 10 : A number is divisible by ten (10) only when its unit digit is zero (0).
Divisibility by 11 : A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places is either zero (0) or number is divisible by 11. In simple words, Add all the digits in the ODD position and all digits in the EVEN position and subtract the smaller result from the larger result. If we get 0 or 11 or any multiples of 11 , then the number is divisible by 11.
Example : 7352631
Sum of odd digits : 7+5+6+1 = 19
Sum of Even digits : 3+2+3 = 8
19-8 = 11. So, number 7352631 is divisible by 11
Divisibility by 12 : A number is divisible by 12 if it is divisible by both three (3) and four (4). In other words, All numbers divisible by both 3 and 4 are divisible by 12.
Divisibility by 15 : A number is divisible by 15 if it is divisible by both three (3) and five (5). In other words, All numbers divisible by both 3 and 5 are divisible by 15.
Divisibility by 25 : A number is divisible by twenty-five (25) only when the number formed by its last two (2) digits is divisible by twenty-five (25).
Divisibility by 27 : A number is divisible by twenty-seven(27) if sum of its digits is divisible by twenty-seven(27).
Divisibility by 125 : A number is divisible by one hundred and twenty-five (125) only when the number formed by its last three (3) digits is divisible by one hundred and twenty-five (125).

An important note : If a number N is divisible by two numbers m and n, where m and n are co-primes(numbers having only 1 as a common factor or Two natural numbers a and b are said to be co-prime if their HCF(Highest Common Factor) is 1.), then N is divisible by mn.

Too err is human. If you see a typo, a spelling mistake, an error, or any other issue, please express via comments.

Speed And Correct Division by 9 (Nine)

2010-12-13T13:08:00.000+05:30

In Vedic Mathematics, there are several divisibility test for different numbers. I am going to explain one of the very simple and correct trick (method) of division by 9. You only require to have knowledge of addition and you will get your Remainder and Quotient. So, Lets start.

Example 1: So, start with easier two digit example. Divide 34 with 9

1. Write extreme Left side digit at bottom.

2. Put the first digit of the dividend as it is under the horizontal line. Put the same digit under the right hand part for the remainder, add the two and place the sum i.e.,, sum of the digits of the numbers as the remainder.

Let's consider some examples of two digits: Find 15/9 , 44/9 and 71/9

1 / 5       4 / 4           7 / 1
     1             4                7
1 / 6        4 / 8           7 / 8

15 / 9 gives Q = 1, R = 6
44 / 9 gives Q = 4, R = 8
71 / 9 gives Q = 7, R = 8

In the division of two digit numbers by 9, we can take the first digit down for the quotient-column and by adding the quotient to the second digit, we get the remainder.

Let's consider some examples of three digits: Find 124 /9, 215 / 9 and 403 / 9

12 / 4       21 / 5           40 / 3
  1 / 3        2 / 3             4 / 4
13 / 7      23 / 8           44 / 7

124 / 9 gives Q = 13, R = 7
215 / 9 gives Q = 23, R = 8
403 / 9 gives Q = 44, R = 7

Note : The remainder is the sum of the digits (digital sum) of the dividend. The first digit of the dividend from left is added to the second digit of the dividend to obtain the second digit of the quotient. This digit added to the third digit sets the remainder. The first digit of the dividend remains as the first digit of the quotient.

Division by 9 (nine) rules:

The remainder is the sum of the digits (digital sum) of the dividend.
The first digit of the dividend from left is added to the second digit of the dividend to obtain the second digit of the quotient.
This digit is added to the third digit set the remainder.

Let's consider some more examples:

Example : Find 611 / 9

Add the first digit 6 to second digit 1 getting 6 + 1 = 7. Hence, Quotient is 67. Now second digit of 67 i.e.,, 7 is added to third digit 1 of dividend to get the remainder i.e.,, 1 + 7 = 8

9 ) 61 / 1
       6 / 7
    67 /  8
Q is 67, R is 8.

Example : Find 234571 / 9

The first digit 2 is set down as the first answer digit . Take this 2 and add the next digit 3. This (2+3) gives 5 as the next digit. Working this way 5+4 =9, 9+5 =14 , 14+7 =21 and the remainder is 21+1 =22

9 ) 234 5 7 / 1
259 ₁4 ₂1/ 22
26061 / 22

The remainder 22 is larger than 9 , the divisor and so divide by 9 giving 2 and remainder 4. This 2 is carried over to the left giving answer are 26063/4

Q= 26063, R = 4

Example : Find 13210 / 9

The first digit 1 is set down as the first answer digit . Take this 1 and add the next digit 3. This gives 4 as the next digit. Working this way 4+2 =6, 6+1 =7 and the remainder is 7+0 =7

9 ) 1321 / 0
     1467 / 7
Q = 1467, R = 7

Find 214091/9

9 ) 2140 9 / 1
2377 ₁6/ 17
23786 / 17

The remainder 17 is larger than 9 , the divisor and so divide by 9 giving 1 and remainder 8. This 1 is carried over to the left giving answer are 23787/8

Q= 23787, R = 8

Too err is human. If you see a typo, a spelling mistake, an error, or any other issue, please express via comments.

Digital roots or Digital Sum of Numbers

2010-12-13T12:40:00.001+05:30

The word Digit means the single figure numbers; the numbers from 1 to 9 and zero

Digital Root or Digital Sum of a number : is the remainder when the number is divided by 9.

So for 31, the remainder is 4 because 31/ 9 = 3 remainder 4. The digital root is also 4.

The digital root can also be obtained by summing the digits of the number.

For example,

Digital sum of 26 is 2 + 6 = 8.
Digital sum of 14 is 1 + 4 = 5.
Digital sum of 764 is 7 + 6 + 4 =17. And 17 is a 2-digit number and we add the figures in 17 to get 1 + 7 = 8. So digital root of 764 is 8.
When the sum of digits is greater than 9, you keep adding. So for 3856, the digital root is 3 + 8 + 5 + 6 = 22, 2 + 2 = 4.

Some facts of digital roots or Digital Sum:

Adding 9 to a number does not affect its digit sum. So 1,10,19, 28 all have a digit sum of 1.
Digital sum of 59309 is 5 + 3 + 0 = 8, ignore all 9's
In terms of digit sums 9 and 0 are equivalent
Any group of digits in a number that add up to a 9 can also be removed.
Digit sum of 26701 is 7.
We see that 2 and 7 which adds up to 9. We can remove 2 and 7 and add up only other digits 6 + 0 + 1 = 7.

If you feel like reading something else, may I recommend:

Speed and correct division by 9 (nine).

To err is human. If you see a typo, a spelling mistake, an error, or any other issue, please express via comments.

How to Multiply with base of 100 – Vedic Multiplication - Lesson Three

2010-05-14T22:55:00.000+05:30

In the last lesson of Vedic Multiplication Methods, I mean here that in my last post I have explained the step-by-step procedure of Vedic Multiplication Methods. If you not read that post, you can read here to understand full concept of this fast calculation methods. In this post, I will demonstrate How to multiply with base of hundred within 15 secs? As I already explained the concept of Vedic Multiplication in my previous post. Now, I will show the examples with little description. But you will be able to understand. So, Lets start with the first example:
How to Multiply with Base of 100
Example 1. Multiply 78 by 96
Step 1. Well, here base is hundred, Put the two numbers 78 and 96 above and below on the left-hand side.
Step 2. For the right-hand-side, find out Deviation of both numbers. That is, 78-100 = -22 and 96-100 = –04 , Write Numbers as shown below:

Step 3. Now, for the right-hand-side of answer multiply the deviations 22 and 04, you will get 88, that is right hand side of the answer.
Step 4. As I already explained that there are four ways of find out the left-hand-side of the answer. Here, I used 78-04=74. You can use any of the four ways to find out left-portion of the answer.

Hence, 78 x 96 = 7488, Here, A vertical dividing line(‘|’) has drawn for the purpose of demarcation of the two parts. Below are the some examples of Multiply with base of 100.

Here are some points to remember:

In all these cases, note that both the cross-subtractions always give the same remainder (for the left-hand-side portion of the-answer)
Here too, note that the vertical multiplication (for the right-hand side portion of the product) may, in some cases, yield a more-than-two-digit product ; but, with 100 as our base, we can have only two digits on the right-hand side. We should therefore adopt the same method as before (i.e. keep the permissible two digits on the right-hand side and "carry" the surplus or extra digit over to the left) (as in the case of ordinary addition, compound addition etc.). See the following examples.

Have you realize that when we multiply with base of 10 and 100 the left portion contains 1 digit and 2 digits respectively. Next, when we multiply with base of 1000 then left side will contain 3 digits. So, fact is that how many zeros('0') a base have, that many digits left side will contain.

Even if you don't understand the above steps. Don't worry! Then, Start from here.

Attempt the following and post their answers in comments:

93x98 = ?
27 x 99 = ?
81x91 = ?
25x78 = ?
56 x98 = ?

So, another lesson of Vedic Multiplication Methods comes to end. In the next lesson we will learn how to multiply with base of 1000.

Multiply 11 to 19 In Your Head

2010-07-11T18:52:00.006+05:30

In this vedic trick you would learn to quickly multiply numbers between 11 and 19 in your head. With this trick, you will be able to multiply any two numbers from 11 to 19 in your head quickly, without the use of a calculator.

It is assumed that you know your multiplication table reasonably well up to 10x10.
Example 1: Multiply 14 by 17

Always place the larger number of the two on top in your mind.
Add right portion of lower number. 17 + 4 = 21
Add a zero behind it (multiply by 10) to get 210.
Multiply right portions of both numbers (4x7 = 28)
Add 210 + 28 = 238.

Hence, 14 x 17 = 238
Example 2: Multiply 15 by 18

Place the larger number of the two on top in your mind.
First add 18 + 5 = 23
Add a zero behind it (multiply by 10) to get 230.
Multiply the covered lower 5 x the single digit above it the "8" (5x8 = 40)
Add 230 + 40 = 270.

Hence, 15 x 18 = 270
That is It! Wasn't that easy? Practice it on paper first!
Note: This Method works only numbers between 11 and 19.

How to Multiply with base of 1000 – Vedic Multiplication – Lesson Four

2010-05-30T12:54:00.002+05:30

What we have done so for? Multiply with base of 100, Multiply with base of 10. Now, we learn how to multiply with base of 1000.
Let’s take first example.
Example 1 .Multiply 787 by 998
Deviation of 998 is –002, and Deviation of 787 is –213.

787 -213

998 -002

(787-002) or (998-213) | (213x2)

Solving above, we get

787 -213

998 -002

785 | 426

Hence, 787x998 = 785426
Let's take another example when right-hand-side contains two digits.
Example 2. Multiply 995 by 987
Deviation of 995 is –005, and Deviation of 987 is –013.

987 -013

995 -005

(987-005) or (995-013) | (13x5)

Solving above, we get

987 -013

995 -005

982 | 065

As I already explained in previous post that how many Zeros(0) a base will have that many digits right side will contain. Here base is 1000, so it will contain 3 digits on the left side.
Hence, 987x995 = 982065
Now, another example when we got carry on right hand side. Example 3 .Multiply 770 by 990
Deviation of 990 is –010, and Deviation of 770 is –230.

770 -230

990 -010

(770-010) or (990-230) | (230x10)

Solving above, we get

770 -230

990 -010

760 | ₂300

Here, we get carry 2, simply add it to left side as we do in natural addition.
Hence, 770x990 = 762300
In this post I just focus on examples. If you not able to understand yet, you can check the previous post. Where I explained full step by step methods.

Current Events 2010

2010-06-16T08:19:00.005+05:30

Friends, You might be thinking that on maths blog where from Current Affairs come. I posted these because SBI is going to recruit 22,000 more clerks at various branches, 'Current Affairs and General Awareness' is main part of their syllabus. It is also helpful for other exams also. So, here are some Current Affairs.

January 2010 Current Events	February 2010 Current Events
World News	World News
U.S. News	U.S. News
Business News	Business News
Disasters and Science News	Disasters and Science News
March 2010 Current Events	April 2010 Current Events
World News	World News
U.S. News	U.S. News
Business News	Business News
Disasters and Science News	Disasters and Science News
May 2010 Current Events	June 2010 Current Events
World News	World News
U.S. News	U.S. News
Business News	Business News
Disasters and Science News	Disasters and Science News

Source: Info Please
If you want to share your knowledge, comment, your experience or even a small problem. Feel free to express via comments. Soon, I'll back with latest news, techniques

Vedic Multiplication Methods – Lesson Two

2010-05-13T00:43:00.001+05:30

Note: This article is related to previous article. If you understand those terms then continue to this article. However, you can continue with this article. But, if you wish to learn from start then, click here.

This method is called, Nikhilam Sutra. Let us first take up a very easy and simple illustrative example (i.e. the multiplication of single-digit numbers above 5) and see how this can be done without previous knowledge of the higher multiplications of the multiplication-tables. Just now, we state and explain the actual procedure, step by step.

Multiplication using a base of 10.

Example 1. Multiply 9 by 8
Consider the base number as 10 since it is near to both the numbers.

Step 1.Put the two numbers 9 and 8 above and below on the left-hand side.

Step 2. Subtract each of them from the base (10) and write down the remainders (1 and 2) on the right-hand side with a connecting minus sign (-) between them, to show that the numbers to be multiplied are both of them less than 10 (as shown below).

Step 3. Now, vertically multiply the two figures (1 and 2). The product is 2. And this is the right-hand-side portion of the answer.

Step 4. Now, the left-hand-side digit (or the answer) can be arrived at in one of 4 ways :--

Cross-subtract in the converse way (i.e. 1 from 8). And you get 7 as the left-hand side portion of the required answer.
8-1=7
Cross-subtract Deviation (2) on the second row from the original number (9) in the first row. And you find that you have got (9-2) i.e. 7 again.
Subtract the sum of the two Deviations (1+2=3) from the base (10). You get the same answer (7) again ; (10-1-2)=7
Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 17). And put (17-10) i.e. 7, as the left-hand part of the answer ;

Hence, 9x8=72
This was easy one and you might also know the answer before start of this example. This method holds good in all cases and is, therefore, capable of infinite application.

Now, time to explain How Nikhilam Sutra Works?

This proves the correctness of the formula. The algebraically explanation for this is very simple :-
(x-a) (x-b) = x (x-a-b)+ab. Where , x is the Base, and a,b are the Deviations respectively. See Proved, below
9 x 8
=(10-1)(10-2)
=10(10-1-2)+(-1)x(-2)
=10(7)+2
=70+2
=72

Lets try another example with different case. A slight difference, however, is noticeable when the vertical multiplication of the right digits (for obtaining the right-hand-side portion of the answer) yields a product consisting of more than one digit. For example, if and when we have to multiply 6 by 7, and write it down as usual :-

7 -3
6 -4
3 /₁2

Here, we notice that the required vertical multiplication (of 3 and 4) gives us the product 12 (which consists of two digits; but, as our base is 10 and the right-hand-most digit is obviously of units, we are entitled only to one digit (on the right-hand side).

This difficulty, however, is easily surmounted with the usual multiplication rule that the surplus portion on the left should always be "carried" over to the left. Therefore, in the present case, we keep the 2 of the 12 on the right hand side and "carry" the 1 over to the left and change the 3 into 4. We thus obtain 42 as the actual product of 7 and 6.

7 -3
6 -4
3 /₁2 = 42

Try this method for solving mathematic problems and beat the clock of Competitive Exams. However, this was easy examples you might know the answers of these. But, In next lessons you will learn how to solve long and complex mathematic calculations within 20 Secs or depends upon your practice.

In the next Lesson, we will learn, How to multiply with base 100

Vedic Multiplication Methods – Lesson One

2010-05-11T00:17:00.001+05:30

Before we start, Let us, understand the following two terms carefully, these will help you to find multiplication in faster and easy way.
Deviation: The difference between the number and the base is termed as deviation. Deviation may be positive or negative. Positive deviation is written without the positive sign and the negative deviation, is written using a bar or negative sign on the number.
Base: Using this multiplication techniques, we relate each number very close to another number called base.
See the following table to understand these terms.

Number	Base	Deviation
18	10	18-10=8
8	10	8-10=-2
95	100	95-100=-5
111	100	111-100=11
994	1000	994-1000=-6
1042	1000	1042-1000=42

So lets start with this easy and fast mathematic technique. There are several cases when we multiply two numbers, listed below:

Both the numbers are lower than the base.
Both the numbers are higher than the base.
One number is higher than the base, and second is lower than the base.
Multiplication using other bases.
Numbers of which the last 2 or 3 or 4 digits added up give 100,1000,10000
Multiplying numbers whose first figures are the same and whose last figures add up to 10, 100 etc.

We will learn these techniques one-by-one. So, these were the basics to start with this technique.

Now, have some water, also take your notepad and pen. To, get start, with these quicker multiplication methods. I hope, you have understand the above terms. So, Don’t get confused in next lessons of this technique. If you don't understand these terms, please review these again and proceed further.

Since, these techniques hard to explain within this single article. So, I split this article into lessons that will help you to learn step-by-step. Continue to second lesson of this article.

Vedic Mathematics Tricks - Multiply by 11 - Part Two

2010-05-09T23:54:00.001+05:30

This is Part II of Multiply by 11, Click Here to read Part I of this post. As you learnt in Part I how to multiply a two digit number with 11. Now, you will learn how to multiply 11 with more than two digits. As usual we deal with simple examples to explain the concept. Here is the first example how to multiply 11 with three digits number.

Example 1. Multiply 145 by 11

Put a Zero(0) on left side so it become 0145

0145

Write down "units" place number, in other words, extreme right digit. On the right side, Here is, ___5
For the "tens" place, add the 4 to the its "neighbor" (the 5). Write the answer : 9 below the 4.
For the "hundreds" place, add the 1 to the its "neighbor" (the 4). Write the answer : 5 below the 1.
For the "thousands" place, add the 1 to the its "neighbor" (the 0). Write the answer : 1 below the 0.

So, what is this technique or you might say that how does it works?

Add a Zero(0) to extreme left.
Write "units" place number digit as it is, on the extreme right side.
Now for "ten", "hundreds", "thousands", "ten-thousands" places and so on. Start adding numbers from right to left.
One more thing when you get "carry" don't forget to add into its "neighbor" on left side.

Note: This technique apply to all digits number to multiply with 11.

2-3 time practice and you will be able to multiply any digit number with 11.

Here are some problems for your practice, post the answers in your comments.

678x11=?
468x11=?
878x11=?
5678x11=?
56759x11=?

If you want to share your knowledge, comment, your experience or even a small problem. Feel free to express via comments. Soon, I'll back with new quick technique.

Vedic Mathematics Tricks - Multiply by 11 - Part One

2010-05-10T00:40:00.001+05:30

Friends, this technique will teaches you how to multiply any number by eleven, easily and quickly. Here, I demonstrated some examples to use this technique. From these examples you will see the pattern used and how easy they are to do.

Example 1. Multiply 45 by 11
Step 1. Split 45 apart, like so: 4__5

4__5

Step 2. Add these two digits together 4+5=9

4+5=9

Final Step. Place the answer, 9 in between 4__5 it gives 495. Which is the correct answer.

495

Was that so simple and quicker than traditional method. Let's try another example:

Example 2. Multiply 68 by 11
Step 1. Again Split 68, like so, 6__8

6__8

Step 2. Adding these two we get, 6+8=14

6+8=14

Final Step. So, we have to put 14 in between 6__8 oh, Don't do this 6148 this is wrong.

Do this : Place 4 from 14 in between 6__8 it will give 648 add 1 as carry to left that 6. Now we get, 748

1

648

748

Here are some problems for your practice, post the answers in your comments.

83x11=?
65x11=?
35x11=?
91x11=?
69x11=?

Continue to second part of this post how to multiply 11 by more than two digits

If you want to share your knowledge, comment, your experience or even a small problem. Feel free to express via comments. Soon, I'll back with new quick technique.