Find the least divisor of 2744 which would make the quotient a perfect square?

Answer 1

It is: 2744/14 = 196
14.

First, factorise the number using power format:

2744 = 2^3 x 7^3

To be a perfect square, each prime must have an even power, so to find out the smallest divisor to use, rewrite each prime with an odd power as itself times the prime to the power of one less:

2744 = (2 x 2^2) x (7 x 7^2)

Now collect the primes with a power of 1 together:

2744 = (2 x 7) x (2^2 x 7^2)

This represents the least divisor: 2 x 7 = 14

since 2^2 x 7^2 = (2 x 7)^2 which is a perfect square:

2744 ÷ 14 = 196 = 14^2