Applications of the Fundamental Theorem of Arithmetic - 1

Finding square numbers.

Let's say we have a number, 234, and we want to know what we need to multiply it by to make a square number.

As a product of prime factors:

234 = 2×3²×13

To make a square number, the powers need to be even.
The power of 3 is already even.
To make the power of 2 even, we need to multiply by another 2.
Similarly for 13.

2×13 = 26

So if we multiply 234 by 26, we will get a square number.

234×26 = 6084

and 6084 = 78²

It is also important to note that:

78 = 2×3×13

Have you noticed the pattern? This is just the prime factors of 6084, but halving each of the powers.

Can you work out what you need to multiply 112 by to make a square number? 

What about 7425?

Can you find a similar pattern for making cubic numbers?