What smallest number should be multiplied to given number so that product is a perfect square?

The following steps will be useful to find the least number which has to multiplied by the given number to get a perfect square.

1. Decompose the given numbers into its prime factors.

2. Write the prime factors as pairs such that each pair has two same prime factors.

3. Find the prime factor which does not occur in pair. That is the least number to be multiplied by the given number to get a perfect square.

Example 1 :

Find the least number multiplied by 200 to get a perfect square.

Solution :

Decompose 200 into its prime factors.

Prime factors of 200 :

200 = 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5

= (2 ⋅ 2) ⋅ 2 ⋅ (5 ⋅ 5)

The prime factor 2 does not occur in pair.

So, '2' is the least number to be multiplied by 200 to get a perfect square.

Justification :

√[2(200)] = √[2(2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5)]

√400 = √[(2 ⋅ 2)(2 ⋅ 2)(5 ⋅ 5)]

= 2 ⋅ 2 ⋅ 5

= 20

Further,

2(200) = 400 = 202

Example 2 :

Find the least number multiplied by 252 to get a perfect square.

Solution :

Decompose 252 into its prime factors.

Prime factors of 252 :

252 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 7

= (2 ⋅ 2) ⋅ (3 ⋅ 3) ⋅ 7

The prime factor 7 does not occur in pair.

So, '7' is the least number to be multiplied by 252 to get a perfect square.

Justification :

√[7(252)] = √[7(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 7)]

√1764 = √[(2 ⋅ 2)(3 ⋅ 3)(7 ⋅ 7)]

= 2 ⋅ 3 ⋅ 7

= 42

Further,

7(252) = 1764 = 422

Example 3 :

Find the least number multiplied by 180 to get a perfect square.

Solution :

Decompose 180 into its prime factors.

Prime factors of 180 :

180 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5

= (2 ⋅ 2) ⋅ (3 ⋅ 3) ⋅ 5

The prime factor 5 does not occur in pair.

So, '5' is the least number to be multiplied by 180 to get a perfect square.

Justification :

√[5(180)] = √[5(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5)]

√900 = √[(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]

= 2 ⋅ 3 ⋅ 5

= 30

Further,

5(180) = 900 = 302

Example 4 :

Find the least number multiplied by 90 to get a perfect square.

Solution :

Decompose 90 into its prime factors.

Prime factors of 90 :

90 = 2 ⋅ 3 ⋅ 3 ⋅ 5

= 2 ⋅ (3 ⋅ 3) ⋅ 5

The prime factors 2 and 5 do not occur in pair.

Product of 2 and 5 :

2 ⋅ 5 = 10

So, '10' is the least number to be multiplied by 90 to get a perfect square.

Justification :

√[10(90)] = √[10(2 ⋅ 3 ⋅ 3 ⋅ 5)]

√900 = √[(2 ⋅ 5)(2 ⋅ 3 ⋅ 3 ⋅ 5)]

= √[(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]

= 2 ⋅ 3 ⋅ 5

= 30

Further,

10(90) = 900 = 302

Example 5 :

Find the least number multiplied by 120 to get a perfect square.

Solution :

Decompose 120 into its prime factors.

Prime factors of 120 :

120 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5

= (2 ⋅ 2) ⋅ 2 ⋅ 3 ⋅ 5

The prime factors 2, 3 and 5 do not occur in pair.

Product of 2, 3 and 5 :

2 ⋅ 3 ⋅ 5 = 30

So, '30' is the least number to be multiplied by 120 to get a perfect square.

Justification :

√[30(120)] = √[30(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)]

√3600 = √[(2 ⋅ 3 ⋅ 5)(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)]

= √[(2 ⋅ 2)(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]

= 2 ⋅ 2 ⋅ 3 ⋅ 5

= 60

Further,

30(120) = 3600 = 602

Kindly mail your feedback to 

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Answer

Hint: First, we will find factors of 98 which are given as ‘numbers on multiplying gives the original number’. For example: to get number 15, factors are 3 and 5. On multiplying 3 and 5, we get 15. Then we will select the smallest number from factors obtained and multiply it to 98, to see that number is a perfect square or not. If yes, we will get the answer.

Complete step-by-step answer:

Note: Students generally make mistakes by assuming that 100 is a perfect square near to 98. So, we will find out on multiplying which number we will get an answer to be 100. So, the equation will be $ 98\times x=100 $ . On solving, we get x as $ x=\dfrac{100}{98}=1.020 $ and write this answer. But this number is in decimal form and is a totally incorrect answer. We have to find integer value and not decimal value. So, be careful in this type of problem.

Solution:

We have to find the smallest whole number by which the number should be multiplied so as to get a perfect square number

To get a perfect square, each factor of the given number must be paired.

(i) 252

Hence, prime factor 7 does not have its pair. If 7 gets a pair, then the number becomes a perfect square. Therefore, 252 has to be multiplied by 7 to get a perfect square.

So, perfect square is 252 × 7 = 1764

1764 = 2 × 2 × 3 × 3 × 7 × 7

Thus, √1764 = 2 × 3 × 7 = 42

(ii) 180

Hence, prime factor 5 does not have its pair. If 5 gets a pair, then the number becomes a perfect square. Therefore, 180 has to be multiplied by 5 to get a perfect square.

So, perfect square is 180 × 5 = 900

900 = 2 × 2 × 3 × 3 × 5 × 5

Thus, √900 = 2 × 3 × 5 = 30

(iii) 1008

Hence, prime factor 7 does not have its pair. If 7 gets a pair, then the number becomes a perfect square. Therefore, 1008 has to be multiplied by 7 to get a perfect square.

So, perfect square is 1008 × 7 = 7056

7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7

Thus, √7056 = 2 × 2 × 3 × 7 = 84

(iv) 2028

Hence, prime factor 3 does not have its pair. If 3 gets a pair, then the number becomes a perfect square. Therefore, 2028 has to be multiplied by 3 to get a perfect square.

So, perfect square is 2028 × 3 = 6084

6084 = 2 × 2 × 13 × 13 × 3 × 3

Thus, √6084 = 2 × 13 × 3 = 78

(v) 1458

Hence, prime factor 2 does not have its pair. If 2 gets a pair, then the number becomes a perfect square. Therefore, 1458 has to be multiplied by 2 to get a perfect square.

So, perfect square is 1458 × 2 = 2916

2916 = 3 × 3 × 3 × 3 × 3 × 3 × 2 × 2

Thus, √2916 = 3 × 3 × 3 × 2 = 54

(vi) 768

Hence, prime factor 3 does not have its pair. If 3 gets a pair, then the number becomes a perfect square. Therefore, 768 has to be multiplied by 3 to get a perfect square.

So, perfect square is 768 × 3 = 2304

2304 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3

Thus, √2304 = 2 × 2 × 2 × 2 × 3 = 48

☛ Check: NCERT Solutions for Class 8 Maths Chapter 6

Video Solution:

NCERT Solutions for Class 8 Maths Chapter 6 Exercise 6.3 Question 5

Summary:

For each of the following numbers, (i) 252 (ii) 180 (iii) 1008 (iv) 2028 (v) 1458 (vi) 768 the smallest whole number by which it should be multiplied so as to get a perfect square number and the square root of the square number so obtained are as follows: (i) 7; √1764 = 42 (ii) 5; √900 = 30 (iii) 7; √7056 = 84 (iv) 3; √6084 = 78 (v) 2; √2916 = 54 and (vi) 3; √2304 = 48

☛ Related Questions: