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
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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
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