# FIND THE LEAST NUMBER TO BE DIVIDED TO GET A PERFECT SQUARE

perfect square is a number that can be expressed as the product of an integer by itself or as the second exponent of an integer.

To check the least number to be divided to get a perfect square, we have to follow the steps.

Step 1 :

Decompose the given number into prime factors.

Step 2 :

After grouping the product of factors as pairs, we may find some more elements must be paired.

Those are the values to be eliminated to make it as perfect square.

Problem 1 :

The least number, by which 1470 must be divided to get a number which is a perfect square, is:

a)   5    b)   6    c)   15     d)   30

Solution :

Prime factorization of 1470 is,

1470 = 2 × 3 × 5 × 7 × 7

We don't see pairs for 2, 3 and 5

2 × 3 × 5 = 30

So, 30 must be divided to 1470 to get a number which is perfect square.

So, option (d) is correct.

Problem 2 :

By which least number should 3087 be divided in order to get a perfect square?

Solution :

Prime factorization of 3087 is,

3087 = 3 × 3 × 7 × 7 × 7

3 has pair, 7 also has pair. But there is one more 7. So, 7 to be divided to make 3087 as perfect square.

Problem 3 :

A man plants 15376 apple trees in his garden and arranges them so that there are as many rows as there are apples trees in each row. The number of rows is

(a)  124   (b) 126  (c) 134   (d) 144

Solution :

15376 15376 = √(2 x 2 x 2 x 2 x 31 x 31)

= 2 x 2 x 31

= 124

So, the number of rows is 124.

Problem 4 :

If 52/x = √(169/289), the value of x is

(a) 52   (b) 58   (c) 62   (d) 68

Solution :

52/x = √(169/289)

52/x = √(13⋅13)//17⋅17)

52/x = 13/17

Doing cross multiplication, we get

13x =  52 (17)

x = 52(17)/13

x = 4(17)

x = 68

So, the value of x is 68.

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