Lesson 2, Topic 3
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# Perfect Square

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A perfect square is a whole number whose square root is a whole number.  Examples

9 = 3 x 3…………… $$\scriptsize \sqrt{9}$$ = 3

4 = 2  x 2…………… $$\scriptsize \sqrt{4}$$ = 2

1 = 1 x 1…………… $$\scriptsize \sqrt{1}$$ = 1

16 = 4 x  4…………… $$\scriptsize \sqrt{16}$$= 4

25 = 5 x 5…………… $$\scriptsize \sqrt{25}$$ = 5

36 = 6  x 6…………… $$\scriptsize \sqrt{36}$$= 6

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 etc are perfect squares because their square roots are whole numbers.

Example 1

Find the smallest number by which 27 must be multiplied to make it a perfect square.

Solution

Step 1:  Express 27 as a product of its prime factors.

Step 2: Pick the product of prime factors in pairs of the same digits. Then multiply digits without pairs (stand-alone digits) together. This gives the required number. i.e

‘ 3’  is  stands alone

The required number  =  3

We can now test our answer to see if this number multiplied by 27 will give a perfect square. 27 x 3 = 81. Our answer is correct because 81 = 9 x 9 which is a perfect square.

Example 2

Find the smallest number by which the following numbers must be multiplied to give perfect squares.

a. 240

b. 45

c. 162

Solution

a. 240

Step 1:

240  =  2 x 2 x 2 x 2 x 3 x 5

Step 2:

The required number  = 3 x 5

=  15

b. 45

Step 1:

45 = 3 x 3 x 5

Step 2:  45 =  (3 x 3) x 5 > No pair

Required number  =  5.

c. 162

Step 1:

Step 2:

2 has no pair. The required number  = 2

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