Question

Find the smallest number by which 2925 must be divided to obtain a perfect square. Also, find the square root of the perfect square.

Answer 1

Hint: Prime factorize 2925. All the primes obtained from the prime factorization should have even exponents to be a perfect square. So if the primes have any odd exponent then eliminate that prime. The resulting number will be a perfect square. Find its square root.

Complete step-by-step 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 \times 975 $
975 can be written as three times 325.
 $ 2925 = 3 \times 3 \times 325 $
325 can be written five times 65
 $ 2925 = 3 \times 3 \times 5 \times 65 $
65 can be written as five times 13.
 $
  2925 = 3 \times 3 \times 5 \times 5 \times 13 \\
  2925 = {3^2} \times {5^2} \times {13^1} \\
 $
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 $ {3^2} \times {5^2} = 9 \times 25 = 225 $
Now, find the square root of 225.
 $
  225 = {3^2} \times {5^2} \\
  \sqrt {225} = \sqrt {{3^2} \times {5^2}} = 3 \times 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.

Note: Prime Factorization is finding the prime numbers which are factors which multiplied together will result in the original number. A perfect square is also called a square number. A perfect square is the product of some integer with itself.