Find the smallest number by which each of the following numbers should be divided so as to get a perfect square. Also, find the square root by the square number thus obtained.
questions
(a) 1575
(b)2700
(c) 3645
Answers
Find the smallest number by which each of the following numbers should be divided so as to get a perfect square. Also, find the square root by the square number thus obtained.
a) 1575
Let us first resolve the given number in prime factorization. We get that,
→ 1575 = 5 × 5 × 3 × 3 × 7
Now, make the pairs.
→ 1575 = 5 × 5 × 3 × 3 × 7
→ 1575 = 5² × 3² × 7
As 7 left unpaired, so 7 is the smallest number by which 1575 should be divided to get the perfect square.
→ New number = Given number ÷ 7
→ New number = 1575 ÷ 7
→ New number = 225
Square root of 225 :
By prime factorization,
→ 225 = 5 × 5 × 3 × 3
→ 225 = 5² × 3²
→ √225 = √(5² × 3²)
→ √225 = √5² × √3²
[ As √(a² × b²) = √a² × √b² ]
→ √225 = 5 × 3
→ √225 =15
So, the square root of the number thus obtained is 15.
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b) 2700
Let us first resolve the given number in prime factorization. We get that,
→ 2700 = 3 × 3 × 3 × 10 × 10
Now, make the pairs.
→ 2700 = 3 × 3 × 3 × 10 × 10
→ 2700 = 3² × 10² × 3
As 3 left unpaired, so 3 is the smallest number by which 2700 should be divided to get the perfect square.
→ New number = Given number ÷ 3
→ New number = 2700 ÷ 3
→ New number = 900
Square root of 900 :
→ 900 = 9 × 100
→ √900 = √9 × √100
→ √900 = 3 × 10
→ √900 = 30
So, the square root of the number thus obtained is 30.
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c) 3645
Let us first resolve the given number in prime factorization. We get that,
→ 3645 = 3 × 3 × 9 × 9 × 5
Now, make the pairs.
→ 3645 = 3 × 3 × 9 × 9 × 5
→ 3645 = 3² × 9² × 5
As 5 left unpaired, so 5 is the smallest number by which 3645 should be divided to get the perfect square.
→ New number = Given number ÷ 5
→ New number = 3645 ÷ 3
→ New number = 729
Square root of 729 :
By prime factorization :
→ 729 = 3 × 3 × 9 × 9
→ 729 = 3² × 9²
→ √729 = √3² × √9²
[ As √(a² × b²) = √a² × √b² ]
→ √729 = 3 × 9
→ √729 = 27
So, the square root of the number thus obtained is 27.