Competitive exams have at least one question where you have to find the square root of a number in seconds. With the help of this shortcut method, you can find square root of a number within seconds. This method is applicable only for perfect squares. The Square root questions asked in competitive exams/bank exams & entrance 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.
math tricks
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.
Sum of Consecutive Numbers Shortcut
Finding the sum of consecutive numbers is a common question asked in competitive exams. You can now find the sum of consecutive numbers starting with 1, the sum of consecutive odd numbers, and the sum of consecutive even numbers in seconds.
What are consecutive numbers?
Numbers that follow each other in order are Consecutive numbers. Here the difference between every two numbers is 1. The consecutive numbers can be represented as n,n+1,n+2.
Examples of consecutive numbers:
(1)1,2,3,4,5,6,7,8
(2)6,7,8,9,10,11
Sum of consecutive numbers starting from 1 shortcut
Now that we know what consecutive numbers are. We shall see a shortcut to find the sum of consecutive numbers starting from 1.
It can be represented as sum of n consecutive numbers = n+(n+1)+(n+2)
Sum of consecutive numbers starting from 1 shortcut steps
Step 1:Count the total number of integers from 1 to the last consecutive number given.
Step 2:Multiply the result in step 1 by 1 more than the number.
Step 3:Divide the result of step 2 by 2.
We shall see some examples to understand better
Example 1:Find the sum of all consecutive numbers from 1 to 9.
Step 1: Counting the number of integers from 1 to 9. We find that there are 9 integers.
Step2: Multiplying the number of integers(9) by 1 more than the number that is 10.
We get, 9 X 10=90
Step3: Dividing the result obtained in step2 by 2
we get, 90/2=45
Ans Hence, the sum of consecutive numbers from 1 to 9=45
Example 2:Find the sum of all consecutive numbers from 1 to 99.
Step 1: Counting the number of integers from 1 to 99. We find that there are 99 integers.
Step2: Multiplying the number of integers(99) by 1 more than the number that is 100.
We get, 99 X 100=9900
Step3: Dividing the result obtained in step2 by 2
we get, 9900/2=4950
Ans Hence, sum of consecutive numbers from 1 to 99=4,950
Odd Consecutive numbers
Odd numbers that follow each other are consecutive odd numbers. Here the difference between every two odd numbers is 2.
It can be represented as Sum of n numbers = n+(n+2)+(n+4)+(n+6) where n is an odd number..
We shall see how to add odd consecutive numbers.
Sum of consecutive odd numbers starting from 1 steps
Step 1: Count the total number of odd integers from 1 to the last consecutive number given.
Step 2: Square the result obtained in step 1.
Example 1:Find the sum of all consecutive odd numbers from 1 to 10.
Step 1: Counting the number of integers from 1 to 9. We find that there are 5 integers.
Step 2: Squaring the result obtained in step 1.
We get, 52=25
Ans hence, Sum of odd numbers from 1 to 9 is 25
Example 2:Find the sum of all consecutive odd numbers from 1 to 100.
Step 1: Counting the number of integers from 1 to 100. We find that there are 50 integers.
Step 2: Squaring the result obtained in step 1.
We get, 502=2500
Ans 2500
Even Consecutive numbers
Even numbers that follow each other are consecutive even numbers. Here the difference between every two even numbers is 2.
It can be represented as Sum of n numbers = n+(n+2)+(n+4)+(n+6)where n is an even number.
We shall see how to add even consecutive numbers.
Sum of consecutive even numbers starting from 2 steps
Step 1: Count the total number of odd integers from 1 to the last consecutive number given.
Step 2: Multiply the result in step 1 by 1 more than the number.
Example 1:Find the sum of all consecutive even numbers from 1 to 10.
Step 1: Counting the number of integers from 1 to 10. We find that there are 5 integers.
Step 2: Multiplying the number of integers(5) by 1 more than the number that is 6.
We get, 5 X 6=30
Ans 30
Example 2:Find the sum of all consecutive even numbers from 1 to 100.
Step 1: Counting the number of integers from 1 to 100. We find that there are 50 integers.
Step 2: Multiplying the number of integers(50) by 1 more than the number that is 51.
We get, 50 X 51=2550
Ans 2550.
2 digit Multiplication Trick with Same Tens digit
You must be wondering we have learned so many multiplication tricks for 2 digit numbers, so how is this going to be helpful. Well, Knowing new tricks and where the trick can be applied to get the answer quickly can make all the difference. 2 digit multiplication trick or double digit multiplication trick whose tens digit is the same is one of those tricks that we will explore today.
Shortcut to find Square Root
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.
Step 2: 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
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If the number ends in
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Unit’s place digit of the square root
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1
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1 or 9(10-1)
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4
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2 or 8(10-2)
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9
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3 or 7(10-3)
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6
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4or 6(10-4)
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5
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5
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0
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0
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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 92 is 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
How to find square root shortcut
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
Also, read Square root shortcut Non-perfect squares