FIND THE LEAST NUMBER TO BE DIVIDED TO GET A 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.

Problem 5 :

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

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

Solution :

To find by which number 1470 must be divided, we write 1470 as product of prime factors.

= √1470

= √5 x 2 x 7 x 7 x 3

Since we have pair of 7, we may factor that out.

After factoring out 7, we get

= 7√(5 x 2 x 3)

There is no pair for 5, 2 and 3. 

= 7√(5 x 2 x 3 x 5 x 2 x 3)

5 x 2 x 3 is needed to be ignored to make it as perfect square.

So, 30 is the required number. Option d is correct.

Problem 6 :

By what least number should 735 be multiplied so that the product is a perfect square. Find the square root of the product so obtained.

Solution 

By writing 735 as product of prime factors,

= √735

= √5 x 7 x 7 x 3)

Factoring out one value for every two same values.

= 7√(5 x 3)

= 7 √(5 x 3 x 5 x 3)

So, 5 x 3 which is 15 should be multiplied to make it as perfect square.

Problem 7 :

A general wishing to draw his 64029 men in the form of a square found that he had 20 men extra. Find the number of men in the front row.

Solution :

From the given information, it is clear that 20 men extra.

64029 - 20 ==> 64009

= √64009

= √253 x 253

= 253

So, the number of men in the front row is 253.

Problem 8 :

Simplify :

√0.0441/√0.000441

Solution :

= √0.0441/√0.000441

To make 0.0441 as fraction, we have to multiply both numerator and denominator by 10000.

= √0.0441 x (10000/10000)

= √(441/10000)

= √(21 x 21)/(100 x 100)

= 21/100

To make 0.000441 as fraction, we have to multiply both numerator and denominator by 1000000.

= √0.000441 x (1000000/1000000)

= √(441/1000000)

= √(21 x 21)/(1000 x 1000)

= 21/1000

= (21/100) / (21/1000)

= 21/100 x 1000/21

= 10

So, the abnswer is 10.

Problem 9 :

The smallest number to be multiplied by 72 to make it a perfect square is

a) 2      b) 3      c) 5

Solution :

= √72

= √(2 x 2 x 2 x 3 x 3)

Here one more 2 is needed to make it as perfect square.

So, option a is correct answer.

Problem 10 :

The smallest number to be divided by 125 to make it a perfect square is

a) 25       b) 5      c) 1

Solution :

= √125

= √(5 x 5 x 5)

Here one more 5 is needed to make it as perfect square.

So, option b is correct.

Problem 11 :

The smallest number to be multiplied by 292 to make it a perfect square is

a) 2      b) 3      c) 73

Solution :

= √292

= √(2 x 2 x 73)

So, 73 is the required number to be multiplied to make it as perfect square.

Problem 12 :

The least number to be divided by 52 to make it a perfect square is

a) 13      b) 4     c) 2

Solution :

= √52

= √(2 x 2 x 13)

So, 13 is the required number to be multiplied to make it as perfect square. Option a is correct.