Question

What is the smallest number by which 2925 must be divided so that the quotient is a perfect cube?with explanation. ​

Answer 1

Answer:

We are given a number 2925. We have to find the number by which 2925 must be divided to obtain a perfect square.

Firstly, prime factorize 2925.

2925 is an odd number so it is not divisible by 2. So start from the next prime which is 3.

2925 can be written as three times 975.

2925=3×975

975 can be written as three times 325.

2925=3×3×325

325 can be written five times 65

2925=3×3×5×65

65 can be written as five times 13.

2925=3×3×5×5×132925=32×52×131

In the result, the exponent of 3 is 2 (even), the exponent of 5 is 2 (even) but the exponent of 13 is 1 which is odd. For a number to be a perfect square the primes in its prime factorization should have even exponents. So eliminate 13.

Therefore, 2925 must be divided by 13 to be a perfect square.

The perfect square will be 32×52=9×25=225

Now, find the square root of 225.

225=32×52225−−−√=32×52−−−−−−√=3×5=15

15 is the square root of 225.

2925 must be divided by 13 to obtain a perfect square 225 and the square root of 225 is 15.