## Shortcut to find Square Root of any Number

In every competitive exam, there is at least one instance where you will have to find the square root of a number quickly. With the help of this shortcut on how to find the square root of a number, you will be able to find out the square root of any number within seconds. This method is applicable only for perfect squares. Square root questions asked in competitive exams/bank exams are usually perfect squares. Refer How to check if a number is a perfect square? to find out if a given number is a perfect square.

### Steps to Find the Square Root of a Number

Step 1: First of all group the number in pairs of 2 starting from the right.

Step2: To get the ten’s place digit, Find the nearest square (equivalent or greater than or less than) to the first grouped pair from left and put the square root of the square.
Step 3: To get the unit’s place digit of the square root
Remember the following
 If the number ends in Unit’s place digit of the square root 1 1 or 9(10-1) 4 2 or 8(10-2) 9 3 or 7(10-3) 6 4or 6(10-4) 5 5 0 0
Let’s see the logic behind this method to find square root for a better understanding
We know,
12=1
22=4
32=9
42=16
52=25
62=36
72=49
82=64
92=81
102=100
Now, observe the unit’s place digit of all the squares.
Do you find anything common?
We notice that,
Unit’s place digit of both 12and 9is 1.
Unit’s place digit of both 22 and 82 is 4
And Unit’s place digit of both 32 and 72 is 9
Unit’s place digit of both 42 and 62 is 6.

Step 4: Multiply the ten’s place digit (found in step 1) with its consecutive number and compare the result obtained with the first pair of the original number from left.
Remember,
If the first pair of the original number > Result obtained on multiplication then select the greater number out of the two numbers as the unit’s place digit of the square root.
If the first pair of the original number < the result obtained on multiplication, then select the lesser number out of the two numbers as the unit’s place digit of the square root.
Let us consider an example to get a better understanding of the method

### Shortcut to find the square root of a 3 Digit Number

#### Example 1: √784=?

Step 1: We start by grouping the numbers in pairs of two from the right as follows
7 84

Step 2: To get the ten’s place digit,
We find that nearest square to the first group (7) is 4 and √4=2
Therefore, ten’s place digit=2.

Step 3: To get the unit’s place digit,
We notice that the number ends with 4, So the unit’s place digit of the square root should be either 2 or 8(Refer table).

Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(2) and its consecutive number(3) we get,
2 x 3=6

ten’s place digit of original number > Multiplication result

7>6

So we need to select the greater number (8) as the unit’s place digit of the square root.

Unit’s place digit =8

Ans:√784=28

### Shortcut to find the square root of any 4 digit number

#### Example 2: √3721?

Step 1: We start by grouping the numbers in pairs of two from the right as follows
37 21

Step 2: To get the ten’s place digit,
We find that nearest square to the first group (37)is 36 and√36=6
Therefore ten’s place digit=6

Step 3: To get the unit’s place digit,
We notice that the number ends with 1, So the unit’s place digit of the square root should be either 1 or 9(Refer table).

Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(6) and its consecutive number(7) we get,
6 x 7=42

ten’s place digit of an original number<Multiplication result

37 < 42

So we need to select the lesser number (1) as the unit’s place digit of the square root.

Unit’s place digit =1

Ans:√3721=61

#### Example 3: √6889?

Step 1: We start by grouping the numbers in pairs of two from the right as follows
68 89

Step 2: To get the ten’s place digit,
We find that nearest square to the first group (68)is 64 and√64=8
Therefore ten’s place digit=8

Step 3: To get the unit’s place digit,
We notice that the number ends with 9, So the unit’s place digit of the square root should be either 3 or 7(Refer table).

Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(8) and its consecutive number(9) we get,
8 x 9 =72

ten’s place digit of an original number<Multiplication result

68 < 72

So we need to select the lesser number (3) as the unit’s place digit of the square root.

Unit’s place digit =3

Ans:√6889=83

### Shortcut to find the square root of any 5 digit number

#### Example 4: √64516

Step 1: We start by grouping the numbers in pairs of two from the right as follows
645 16

Step 2: To get the ten’s place digit,
We find that nearest square to the first group (645)is 625 and√625=25
Therefore ten’s place digit=25

Step 3: To get the unit’s place digit,
We notice that the number ends with 6, So the unit’s place digit of the square root should be either 4 or 6(Refer table).

Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(25)and its consecutive number(26)we get,
25 x 26=650

ten’s place digit of an original number<Multiplication result

645< 650

So we need to select the lesser number (4) as the unit’s place digit of the square root.

Unit’s place digit =4

Ans:√64516=254

### Shortcut to find the square root of any 6 digit number

#### Example 5: √126736

Step 1: We start by grouping the numbers in pairs of two from the right as follows. For a 6 digit number, this is how it should be paired.
1267 36

Step 2: To get the ten’s place digit,
We find that nearest square to the first group (1267)is 1225 and√1225=35
Therefore ten’s place digit=35

Step 3: To get the unit’s place digit,
We notice that the number ends with 6, So the unit’s place digit of the square root should be either 4 or 6(Refer table).

Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(35)and its consecutive number(36)we get,
35 x 36=1260

ten’s place digit of original number>Multiplication result

1267>1260

So we need to select the greater number (6) as the unit’s place digit of the square root.

Unit’s place digit =6

Ans:√126736=356

#### Example 6: √207025

Step 1: We start by grouping the numbers in pairs of two from right as follows. For a 6 digit number, this is how it should be paired.
2070 25

Step 2: To get the tenth place digit,
We find that nearest square to the first group (2070) is 2025 and√2025=45. Therefore ten’s place digit=45

Step 3: To get the unit’s place digit,
We notice that the number ends with 5, So the unit’s place digit of the square root should be 5(Refer table).
Skip all the other steps for a number ending in 5

Unit’s place digit =5

Ans:√207025=455

## Divide number by 15 Shortcut

Dividing can be a time-consuming task. However, if you know how to divide number by 15 shortcut, you bet, you can do it in seconds. Without much adeau, let us see the steps on how it can be done.

Below are the steps
Step 1: Move the decimal point one place to the left of the number.
Step 2: Now, double the number
Step 3: Divide the result obtained in step 2 by 3 to get your answer.

### Examples of how to divide number by 15

Example 1: Divide 221 by 15
Step 1: Moving the decimal point to the left of the number we get 22.1
Step 2: Doubling
we get, 22.1 x 2=44.2
Step 3: Dividing the result obtained in step 2 by 3
we get, 44.2/3= 14.73

Ans Thus, 221/15= 14.73

Example 2: Divide 256 by 15
Step 1: Moving the decimal point to the left of the number we get 25.6
Step 2: Doubling
we get, 25.6 x 2= 51.2
Step 3: Dividing the result obtained in step 2 by 3
we get, 51.2/3= 17.06

Ans Thus, 256/15= 17.06

Example 3: Divide 14 by 15
Step 1: Moving the decimal point to the left of the number we get 1.4
Step 2: Doubling
we get, 1.4 x 2= 2.8
Step 3: Dividing the result obtained in step 2 by 3
we get, 2.8/3= 0.93

Ans Thus, 14/15= 0.93

Example 4: Divide 22 by 15
Step 1: Moving the decimal point to the left of the number we get 2.2
Step 2: Doubling
we get, 2.2 x 2= 4.4
Step 3: Dividing the result obtained in step 2 by 3
we get, 4.4/3= 1.46

Ans Thus, 22/15= 1.46

Example 5: Divide 1024 by 15
Step 1: Moving the decimal point to the left of the number we get 102.4
Step 2: Doubling
we get, 102.4 x 2= 204.8
Step 3: Dividing the result obtained in step 2 by 3
we get, 204.8/3= 68.26

Ans Thus, 1024/15= 68.26

## Divisibility Rules| Number divisible by 2,3,4,5,6,7,8,9,10,11 or 13

Divisibility Rules are useful when you want to find out if the given big number is exactly divisible or not without actually dividing.
This saves a lot of time which is the need of the hour in any competitive exam. This shortcut is especially handy when you are solving Problems on Ages. We shall find out if a number is divisible by 2,3,4,5,6,7,8,9,10,11,13 using various divisibility rules.

### Divisibility Rules | Number divisible by 2

To check if the given number is exactly divisible by 2 follow the below steps
Step 1: Check if the units digit of the given number is even.
Step 2: If the units digit is even, the given number is exactly divisible by 2.
If not, the given number is not exactly divisible by 2.

Example 1:Is 584 exactly divisible by 2.
Step 1: We see that the units digit (4)is an even number.
Thus, we can say that 584 is exactly divisible by 2.

Example 2: Is 627 exactly divisible by 2.
Step 1: We see that the units digit (7)is an odd number.
Therefore, we can conclude that 627 is not exactly divisible by 2.

### Divisibility Rules | Number divisible by 3

To check if the given number is exactly divisible by 3 follow the below steps
Step 1: Add all the digits in the given number until you arrive at a single number.
Step 2: If the single number arrived is 3, 6, 9, then, the given number is exactly divisible by 3. If not the given number is not exactly divisible by 3.

Example 1: Is 95 exactly divisible by 3.
Adding we get, 9 + 5 = 14
Further adding we get,1 + 4=5
we know that 5 is not exactly divisible by 3.
Thus, we can conclude that 95 is not exactly divisible by 3.

Example 2: Is 63 exactly divisible by 3
Adding we get, 6 + 3=9
We know that 9 is exactly divisible by 3.
Therefore, 63 is exactly divisible by 3.

### Divisibility Rules | Number divisible by 4

To check if the given number is exactly divisible by 4 follow the below steps
Step 1: Check if the last two digits of the given number are divisible by 4.
Step 2: If divisible then, the entire number is divisible by 4.
If not, then the entire number is not divisible by 4.

Example 1: Is 624 divisible by 4?
Step 1: Dividing last 2 digits(24) by 4.
We know that 24 is exactly divisible by 4.
So, we can conclude that the given number 624 is also exactly divisible by 4.

Example 2: Is 514 divisible by 4?
Step 1: Dividing last 2 digits(14) by 4.
We know that 14 is not exactly divisible by 4.
So, we can conclude that the given number 514 is also not exactly divisible by 4.

### Divisibility Rules | Number divisible by 5

To check if the given number is exactly divisible by 5 follow the below steps
Step 1: If the given number ends in 0 or 5 it is exactly divisible by 5.
If not then, the given number is not exactly divisible by 5.

Example 1: Is 1015 divisible by 5?
Step 1: We see that the given number ends in 5
Thus, we can say that 1015 is exactly divisible by 5.

Example 2: Is 5551 divisible by 5?
Step 1: We see that the given number ends in 1
So, we can conclude that 5551 is not exactly divisible by 5.

### Divisibility Rules | Number divisible by 6

To check if the given number is exactly divisible by 6 follow the below steps
Step 1:
Check if the given number is Even number.
Step 2: Find the sum of all the digits of the number until you arrive at a single number.
Step 3: If the single number arrived is 3, 6, 9, then, the given number is exactly divisible by 6. If not the given number is not exactly divisible by 6.

Example 1: Is 846 divisible by 6?
Step
1: We see that the given number is an even number.
Step 2: Add the digits we get, 8 + 4 + 6=18
Again adding we get, 1 + 8= 9
Therefore, the given number 846 is exactly divisible by 6.

Example 2: Is 825 divisible by 6?
Step 1: We see that the given number is an odd number.
Step 2: Add the digits we get,8 + 2 + 5=15
Further adding we get, 1 + 5 =6
Even though the sum of the digits is 6, the given number is not exactly divisible by 6 as the given number is odd number.

### Divisibility Rules | Number divisible by 7

To check if the given number is exactly divisible by 7 follow the below steps
Step 1: Multiply the last digit of the given number by 2.
Step 2: Subtract the result obtained in step 1 from the remaining digits of the given number.
Step 3: If the result obtained in Step 2 is either 0 or a number divisible by 7 then, the given number is exactly divisible by 7. If not then, the given number is not exactly divisible by 7.

Example 1: Is 259 divisible by 7?
Step 1: Multiplying the last digit by 2 we get, 9 X 2=18
Step 2: Subtracting the result we get, 25 – 18= 7
Step 3: We know that 7 is exactly divisible by 7
So we can conclude that 259 is exactly divisible by 7.

Example 2: Is 123 divisible by 7?
Step 1: Multiplying the last digit by 2 we get, 3 X 2=6
Step 2: Subtracting the result we get, 12 – 6= 6
Step 3: We know that 6 is not exactly divisible by 7
Thus, we can say that 123 is not exactly divisible by 7.

### Divisibility Rules | Number divisible by 8

To check if the given number is exactly divisible by 8 follow the below steps
Step 1: Check if the given number is Even number.
Step 2: Check if the last 3 digits of the given number is divisible by 8.
Step 3: If divisible then, it is exactly divisible by 8. If not then, it is not exactly divisible by 8.

Example 1: Is 2016 divisible by 8?
Step 1: We see that the given number is an even number.
Step 2: The last 3 digit that is 016 is exactly divisible by 8.
Step 3: Thus, we can conclude that the given number 2016 is exactly divisible by 8.

Example 2: Is 2124 divisible by 8?
Step 1: We see that the given number is an even number.
Step 2: The last 3 digits that is 124 is not exactly divisible by 8.
Step 3: So, we can say that the given number 2124 is not exactly divisible by 8.

### Divisibility Rules | Number divisible by 9

To check if the given number is exactly divisible by 9 follow the below steps
Step 1: Add all the digits of the number.
Step 2: If the number obtained is 9 or a multiple of 9, then the given number is exactly divisible by 9.
If not then, the given number is not exactly divisible by 9.

Example 1: Is 4513 divisible by 9?
Step 1: Adding the digits in the given number we get, 4+5+1+3=13
Step 2: We know that 13 is not divisible by 9.
Step 3
: Thus, we can say that the given number is not exactly divisible by 9.

Example 2: Is 3555 divisible by 9?
Step 1: Adding the digits in the given number we get, 3+5+5+5=18
Step 2: We know that 18 is exactly divisible by 9.
Step 3: Thus, we can say that the given number is exactly divisible by 9.

### Divisibility Rules | Number divisible by 10

To check if the given number is exactly divisible by 10 follow the below steps
Step 1: If the last digit of the given number is 0 then it is exactly divisible by 10.
Step 2: If not then, the given number is not exactly divisible by 10.

Example 1: Is 5510 divisible by 10?
Step 1: Here, we see that the last digit of the given number is 0.
Thus, we can say that the given number is divisible by 10.

Example 2: Is 6201 divisible by 10?
Step 1: Here, we see that the last digit of the given number is not 0.
Thus, we can say that the given number is not exactly divisible by 10.

### Divisibility Rules | Number divisible by 11

To check if the given number is exactly divisible by 11 follow the below steps
Step 1: Find alternative digits in the given number and make 2 separate groups.
Step 2: Find the sum of the digits within each group.
Step 3: Find the difference between the 2 sums.
Step 4: If the difference in the sums is equal to 0 or 11 or multiple of 11, then the given number is exactly divisible by 11.
If not then, the given number is not exactly divisible by 11.

Example 1: Is 5962 divisible by 11?
Step 1: Finding alternative digits and grouping
we get, 5 and 6 -group A and 9 and 2- group B
Step 2: Finding sum of the digits
we get, Group A- 5+6=11 and Group B- 9+2=11
Step 3: Finding the difference between group A and group B
we get, 11-11=0
Step 4: We see that the difference is 0. Therefore, we can say that the given number 5962 is exactly divisible by 11

Example 2: Is 3431 divisible by 11?
Step 1: Finding alternative digits and grouping
we get, 3 and 3 -group A and 4 and 1- group B
Step 2: Finding sum of the digits
we get, Group A- 3+3=6 and Group B- 4+1=5
Step 3: Finding difference of group A and group B
we get, 6-5=1
Step 4: We see that the difference is not 0 or multiple of 11. Thus, we can say that the given number 3431 is not exactly divisible by 11

### Divisibility Rules | Number divisible by 13

To check if the given number is exactly divisible by 13 follow the below steps Step 1: Find alternative groups of 3 digits from the given number starting from the right. One alternative group to be group A and another alternative group to be Group B.
Step 2: Add the alternative Group A and Group B to arrive at two sums.
Step 3: Find out the difference between the 2 sums.
Step 4: If the difference is equal to 0 or a multiple of 13 then, the given number is exactly divisible by 13.
If not the given number is not exactly divisible by 13.

Example 1: Is 2,456,121,241,514 divisible by 13?
Step 1: Finding alternative groups
We get, Group A- 514,121,2
And Group B- 241,456
Step 2: Adding the alternative Groups
We get, Group A =514+121+2=637
Group B= 241+456=697
Step 3: Finding the difference between the 2 sums.
We get, Group A- Group B- 637-697=60
Step 4: We see that the difference(60) is neither equal to 0 nor a multiple of 13. Thus ,the given number is not exactly divisible by 13.

Example 2: Is 32,903 divisible by 13?
Step 1: Finding alternative groups
We get, Group A-903
And Group B-32
Step 2: Adding alternative Groups.This step can be skipped as there is only 1 number in the group.
We get, Group A=903 and Group B=32
Step 3: Finding the difference between the 2 sums.
We get, Group A- Group B- 903-32=871
Step 4: We see that the difference(871) is a multiple of 13. Thus, the given number is exactly divisible by 13.

## Problems On Ages Shortcuts

Problems on Ages is one of the questions asked in every Bank Exam. When you know that this is a sure question, why not prepare yourself well to stand up to the competition. Equip yourself with this Problems on Ages shortcuts and you can solve problems on ages like a PRO. And if you know divisibility tricks, its cake walk for you. You can use both tricks to solve the problem quickly with accuracy.

Let me walk you through few examples to get an understanding of the shortcut.

### Problems on Ages Examples

#### Example 1: Present ages of S and A are in the ratio of 5:4 respectively. Three years hence, the ratio of their ages will become 11:9 respectively. What is A’s present age in years? A)40 years B)43 years C)24 years D)27 years E)None of the above

Step 1:Write down the condition 1 given in question as follows
Present ages of S & A= 5:4

And we need to find A’s present age.

Step 2: Observe the options given in answer and find out which option is exactly divisible by 4.
We see that,

option A) 40/4=10 Exactly divisible by 4 so this could be the answer.
option B) 43/4 Not exactly divisible by 4
option C) 24/4=6 Exactly divisible by 4 so this could be the answer.
option D) 27/4 Not exactly divisible by 4.

So we have 2 options A and C exactly divisible by 4.

Step 3: Since we have 2 options satisfying condition 1. We need to check the options for condition 2 to see which among them is the answer.

Step 4: Condition 2 is Three years hence, the ratio is 11:9 is given.
First, we shall add 3 years to each of the options to get the age after 3 years(3 years hence)and divide each of the options by 9(A’s part)
We see that,

Option A) 40+3=43 years 43/9= Not exactly divisible by 9.
Option B) 43+3=46 years 46/9= Not exactly divisible by 9.
Option C) 24+3=27 years 27/9= Exactly divisible by 9.
Option B) 27+3=30 years 30/9= Not exactly divisible by 9.

We see that only option C satisfies condition 2.

Therefore, we can say that option C is the correct answer as it satisfies both condition 1 and condition 2.

A’s present age is 24 years.

#### Example 2: Six years ago, the ratio of the ages of K and S was 6:5. Four years hence, the ratio of their ages will be 11:10. What is S’s age at present? A)18 years B)16 years C)20 years D)25 years E)None of the above

Step 1:  Condition1 given in the question is 6 years ago, the ratio was 6:5 is given
First, we shall subtract 6 years from each of the options to get the age before 6 years(6 years ago)and divide each of the options by 5(S’s part)
We see that,

Option A) 18-6=12 years 12/5 Not exactly divisible by 5.
Option B) 16-6=10 years 10/5=5 Exactly divisible by 5.
Option C) 20-6=14 years 14/5 Not exactly divisible by 5.
Option D) 25-6=19 years 19/5 Not exactly divisible by 5.

We see that only option B satisfies condition 1.

Thus, we can say that option C is the correct answer as it is the only option which satisfies the condition1.
We do not have 2 options satisfying condition1, so no need to check if the options satisfy condition 2. We can stop here and save a few seconds here.

S’s present age is 16 years.

## Square Root Shortcut Nonperfect Squares

In the previous trick, we learned how to find the square root of any number which is a perfect square. Now, using this square root shortcut, you can easily find the square root of a nonperfect square. For instance, lets say you want to find √26. Guess what, you can find the square root of this nonperfect square in seconds. Here is how you can do it.

## Square Root Shortcut for Nonperfect squares steps

Step1: Find the nearest square less than or greater than a perfect square and find its square root.
Step 2: Now, find the difference between the nearest square and the given number.
Step 3: Multiply the result obtained in step 1 by 2
Step 4: Divide the result obtained in step 2 by the result obtained in step 3.
Step 5: Combine the results of step 1 and step 4 to arrive at your answer.

Remember the below squares
12=1
22=4
32=9
42=16
52=25
62=36
72=49
82=64
92=81
102=100
112=121
122=144
132=169
142=196
152=225
162=256
172=289
182=324
192=361
202=400

Let us now see a few examples to understand better

## Examples

Example 1: Find √26

Step 1: Finding the nearest square to number 26. We find that 25 is the nearest square to 26. √25 =5

Step 2: Finding the difference between the nearest square and the given number. Here, 26-25= 1. Difference= 1

Step 3: Multiplying the result obtained in step 1 by 2.

We get, 5 X 2=10

Step 4: Dividing the result obtained in step 2 by result obtained in step 3.

We get, 1/10=0.10

Step 5: Combining the results of step 1 and step 4.

We get, 5 + 0.10= 5.10

Ans √26 = 5.10

Example 1: Find √38

Step 1: Finding the nearest square to number 38. We find that 36 is the nearest square to 38. √36 =6

Step 2: Finding the difference between the nearest square and the given number. Here, 38-36= 2. Difference= 2

Step 3: Multiplying the result obtained in step 1 by 2.

We get, 6 X 2=12

Step 4: Dividing the result obtained in step 2 by result obtained in step 3.

We get, 2/12=0.16

Step 5: Combining the results of step 1 and step 4.

We get, 6 + 0.16= 6.16

Ans √38 = 6.16