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How to check if a number is a perfect square?

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How to check if a number is a perfect square?

A number is a perfect square if given number is equal to the square of some natural number.

Using the prime factorization method we can find out if a given number is a perfect square or not. Moreover, by this method you can accurately say whether a number is a perfect square or not. We will not guess the answer as we all know that there is penalty for every wrong answer.
Prime numbers between 1 to 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
Let us now go through the steps
Step 1: Find prime factors of the given number
Step 2 : Group the factors in pairs
Step 3: After grouping if you find that
          → The number is a perfect square if No factor is left
          → Number is not perfect square if Factor/Factors are left
We shall now apply this trick in our example

Examples on How to check if a number is a perfect square

Example 1:Is 1024 a perfect square?
Step 1: Finding prime factors of 1024

verify whether 1024 is a perfect square or not

Step 2: Grouping factors in pairs as shown below 1024=(2×2)x(2×2)x(2×2)x(2×2)x(2×2)

Step 3:After grouping we notice that no factor is left
Therefore, we can conclude that without doubt 1024 is a perfect square.

Example 2: Is 784 a perfect square?
Step 1: Finding prime factors of 784
Is 784 a perfect square?
Step 2: Grouping factors in pairs as shown below784=(2×2)x(2×2)x(7×7)
Step3: After grouping we notice that no factor is left
Therefore, we can conclude that 784 is absolutely a perfect square.
Example 3: Is 3721 a perfect square?
Step 1: Finding prime factors of 3721.
Is 3721 a perfect square?
Step 2: Grouping factors in pairs as shown below
3721=(61×61)
Step 3: After grouping we notice that no factor is left
Therefore, we can conclude that 3721 is absolutely a perfect square.
Example 4:Is 6889 a perfect square?
Step 1: Finding prime factors of 6889.
Is 6889 a perfect square?
Step2: Grouping factors in pairs as shown below
6889=(83 x 83)
Step 3:After grouping we notice that no factor is left
Therefore, we can conclude that 6889 is absolutely a perfect square.
Tip: Remember the squares of prime numbers as it can save a lot of time.

Shortcut to divide 2 digit by 9

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Shortcut to divide any 2 digit number by 9

Dividing a two digit number by 9 using the method taught to us in our school days may take a minute. However, by applying the shortcut to divide 2 digit by 9 you can mentally calculate the answer within seconds.

Shortcut to divide 2 digit by 9 steps

Let us see the steps now
Step 1: Enter the ten’s digit of the number as it is.
Step 2: To get the unit’s place digit add the ten’s place digit number and Unit’s place digit number and put the decimal point after the ten’s place digit.

Examples

Let us consider an example
Example 1:23÷ 9=?
Step 1: We enter the ten’s place digit as it is
Ten’s place digit=2
Step 2: We get the unit’s place digit by adding ten’s place digit and unit’s place digit
2+3=5
Unit’s place digit=5
Now add a decimal point between the unit’s place digit and the ten’s place digit to arrive at the answer
Ans          23÷ 9 =2.5
Example 2: 32÷ 9=?
Step 1: We enter the ten’s place digit as it is
Ten’s place digit=3
Step 2: We get the unit’s place digit by adding ten’s place digit and unit’s place digit
3+2=5
Unit’s place digit=5
Now add a decimal point between the unit’s place digit and the ten’s place digit to arrive at the answer
Ans          32÷ 9 =3.5
Example 3: 47÷ 9=?
Step 1: We enter the ten’s place digit as it is
Ten’s place digit=4
Step 2: We get the unit’s place digit by adding ten’s place digit and unit’s place digit
4+7=11.
Since it is a 2 digit number, add both numbers that is 1 + 1 to get unit’s place digit. We get, unit place digit=1+1=2
Also carry forward the ten’s place digit to step 1
Step 3: Add carry forward(1) from step 2 to original ten’s place digit of given number. we get, ten’s place digit= 4 + 1=5
Now add a decimal point between the unit’s place digit and the ten’s place digit to arrive at the answer
Ans          47÷ 9 =5.2

Shortcut to multiply any 2 digit number by 11

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Shortcut to multiply any 2 digit number by 11
Multiplying any 2 digit number by 11 is the easiest when you know this trick. Shortcut to multiply 2 digit by 11 is very easy to mentally get the correct answer. It saves a lot of time when this is part of a bigger problem given in bank exams.
 

Shortcut to multiply 2 digit by 11 

 Let us now quickly go through the steps
Step 1: Enter the unit’s place digit as it is. This is the last digit of your answer.
Step 2: Enter the ten’s place digit leaving a blank in between for the middle digit.
Step 3: To find the middle digit, add unit’s place digit and ten’s place digit and input the digit in the blank.
NOTE: On adding if you arrive at a 2 digit number then, input only the unit’s place of the middle digit in the blank and add the ten’s place digit of the middle digit to the original ten’s place digit of the number.
Let me explain with an example to make it easy to understand
Example 1: 42 X 11 
Step 1: Here the unit’s place digit is 2, enter as it is i.e 2
Step 2: Here the ten’s place digit is 4, so input it as follows leaving a blank between the unit’s place digit and the ten’s place digit for the middle digit.
4_2
Step 3: Now, By adding unit’s place digit and ten’s place digit (4+2=6)we get the middle digit as 6 which we enter in the blank as follows.
462
Ans : 42 x 11=462
Example 2: 31 X 11 
Step 1: Here the unit’s place digit is 1, enter as it is i.e 1
Step 2: Here the ten’s place digit is 3, so input it as follows leaving a blank between the unit’s place digit and the ten’s place digit for the middle digit.
3_1
Step 3: Now, By adding unit’s place digit and ten’s place digit (3+1=4)we get the middle digit as 4 which we enter in the blank as follows.
341
Ans : 31 x 11=341
Example 3: 53 X 11 
Step 1: Here the unit’s place digit is 3, enter as it is i.e 3
Step 2: Here the ten’s place digit is 5, input it as follows leaving a blank between the unit’s place digit and the ten’s place digit for the middle digit.
5_3
Step 3: Now, By adding unit’s place digit and ten’s place digit (3+5=8)we get the middle digit as 8 which we enter in the blank as follows.
583
Ans : 53 x 11=583
Example 4: 67 X 11 
Step 1: Here the unit’s place digit is 7, enter as it is i.e 7
Step 2: Here the ten’s place digit is 6, so input it as follows leaving a blank between the unit’s place digit and the ten’s place digit for the middle digit.
6_7
Step 3: Now, By adding unit’s place digit and ten’s place digit (7+6=13)we get
2 digit number then, input only the unit’s place of the middle digit(3) in the blank and carry forward the ten’s place digit to step 4.
637
Step 4: Add the ten’s place digit(1) to the original ten’s place digit(6) of the number as follows.
6+1=7
Ans : 67 x 11=737
Example 5: 49 X 11 
Step 1: Here the unit’s place digit is 9, enter as it is i.e 9
Step 2: Here the ten’s place digit is 4, so input it as follows leaving a blank between the unit’s place digit and the ten’s place digit for the middle digit.
4_9
Step 3: Now, By adding unit’s place digit and ten’s place digit (9+4=13)we get
2 digit number then, input only the unit’s place of the middle digit(3) in the blank and carry forward the ten’s place digit to step 4.
439
Step 4: Add the ten’s place digit(1) to the original ten’s place digit(4) of the number as follows.
4+1=5
Ans : 49 x 11=539
Tip to remember: Ten’s place (ten’s place + unit place) Unit’s place

How to verify if a number is perfect cube

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How to verify if a number is perfect cube
Learn how to verify if a number is perfect cube using Prime Factorization Method. It is the method to use to find out if a given number is a perfect cube or not if you want accurate results. Before going into the details of the Prime Factorization Method, it is helpful to know what Prime factors are.

Prime numbers are those numbers which are divisible only by 1 and the number itself. These numbers do not have any other divisible factors.

Prime numbers such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29……. are also known as the prime factors.
Now, we shall see how Prime Factorization Method is done from the below steps.
Step 1: Find the prime factors of the given number.
Step 2:Group the factors in 3’s of equal factors.
Step 3:After grouping,
If you are not left with any separate single or double factor, then the given number is a perfect cube.
If you are left with any separate single or double factor, then the given number is not a perfect cube.
Let us consider some examples to understand the Prime Factorization Method better.

Examples

Example of a number which is a perfect cube.

Example1:Verify if 216 is a perfect cube.
Step1:We start by finding prime factors of 216.
prime factorization1
Prime factors of 216 are 2x2x2x3x3x3
Step 2: Grouping the factors into 3’s
          (2x2x2) x (3x3x3)
Step 3: After grouping, we notice that no factor is left.
Therefore we can say that 216 is definitely a perfect cube.

Example of a number which is not a perfect cube.

Example 2:Verify if 1024 is a perfect cube
Step1:We start by finding prime factors of 1024.
prime factorization
Prime factors of 1024 are 2x2x2x2x2x2x2x2x2x2
Step 2:Grouping the factors into 3’s
(2x2x2) x (2x2x2) x (2x2x2) x 2
Step 3: After grouping, we notice that factor 2 is left.
Therefore we can say that 1024 is not a perfect cube.

Shortcut to find cube of any number from 1 to 100

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Shortcut to find cube of any number from 1 to 100

Finding cube of a number above 20 in the usual way is difficult and time consuming. With this Shortcut to find cube of any number, you can easily calculate cube of a number within seconds.

Short cut to find cube of number steps

Step 1: Assume the ten’s place number of the given number as a and the unit’s place number as b.
Step 2: Now, we all know that (a+b)³=a³ + 3a²b + 3ab² + b³

We shall manipulate the same formula to calculate the cube of a number.

(i)Find b³ to get the last digit
 If you get two digit number then, add ten’s place digit to 3ab²
(ii) Find 3ab²
If you get two digit number then, keep the unit’s place digit and and
add ten’s place digit to 3a²b.

(iii) Find 3a²b
If you get two digit number then, keep the unit’s place digit and and
add ten’s place digit to a³.

(iv) Find a³
If you get two digit number then, just add the carried forward digit if any
and write the number.
Cube of any 2 digit number will be in the pattern of
a³ 3a²b  3ab² b³

Cube To make things easier it is good to memorize the cubes of numbers from
1 to 10
13=1
23=8
33=27
43=64
53=125
63=216
73=343
83=512
93=729

Examples

Lets use this method to calculate the cube of a number with help of an example to understand better

Example 1: 32³=?
Step 1: Assume a= 3 and b= 2
Step 2: Now substituting the values of a and b in the below pattern
a³ 3a²b  3ab² b³

(i) Finding b³  Now that is
b³=2³=8
We get the last digit as 8.

(ii) Finding 3ab²
3ab²=3(3)(2)²=36
From this we will keep the one’s place digit(6) and add the ten’s place digit(3) to 3a²b.

(iii)Finding 3a²b and adding ten’s place digit(3)
3a²b=3(3)²(2)=3(9)(2)=54 + 3= 57

From this we will keep the one’s place digit(7) and add ten’s place digit(5) to b in the next step.

(iv) Finding a³ and adding the ten’s place digit(5) from previous step
a³=3³=27+5=32
From this we get the number as 32

From all the above steps we get

Ans 32³=32768

Example 2: 47³=?
Step 1: Assume a= 4 and b= 7
Step 2: Now substituting the values of a and b in the below pattern
a³ 3a²b  3ab² b³

(i) Finding b³  Now that is
b³=73=343
We get the last digit as 3.

(ii) Finding 3ab²
3ab²=3(4)(7)²=12 x 49=588 (Tip: Use Shortcut to multiply any 2 digit by 2 digit)
Now add 34 that you got from step 1 to 588,we get  588+34=622
From this we will keep the one’s place digit(2) and add remaining digits(62) to 3ab².

(iii)Finding 3a²b and adding remaining digits of step 2(62)
3a²b=3(4)²(7)=16 x 21=336
336+ 62=398
From this we will keep the one’s place digit(8) and add remaining digits(39) to a³.

(iv) Finding a³ and adding the remaining digits from step 3
a³=4³=64
64 + 39=103

From all the above steps we get
47³=103823

With practice you will be able to find the cube of any two digit
number within seconds.

Try it  with other numbers and let us know if it was helpful to you!

If you have any other shortcut way of finding the cube of a number,
feel free to share it here.