# Perfect Roots Puzzle

What is the lowest number (other than the trivial answer of one), which is a perfect square, perfect cube, and perfect fifth power?
Specifically what is the lowest N that satisfies the equation:
N = a2 = b3 = c5
Where a,b,c are whole numbers.

## Trivia Diversion

Interesting piece of trivia: The last digit of any number raised to the fifth power does not change. e.g.:
65 = 7,776
1195 = 23,863,536,599
145 = 537,824
1,0075 = 1,035,493,442,021,807

## Solution

We could brute force this, but we don’t really know what the range is, plus it seems wasteful when we can apply some simple math to get a solution.
First of all, we want to get all the exponents of all the numbers the same.
N = a2 = b3 = c5 = xz
We can represent N as some value x raised to the power z. To find a suitable value for z, we can use the lowest common multiple of the powers 2,3,5. As these are all relatively prime, the answer is 30.
N = x30
Because of the way powers work, we can split this common multiple into the components we need:
N = (x2)15 = (x3)10 = (x5)6
Which of course is equivalent to:
N = (x15)2 = (x10)3 = (x6)5
We want the lowest value, so after the trivial x = 1, we can insert x = 2, which gives the answer: 1,073,741,824
The smallest number, greater than one, which is a perfect square, cube, and fifth power is 1,073,741,824
1,073,741,824 = (215)2 = (210)3 = (26)5
1,073,741,824 = 32,7682 = 1,0243 = 645
This answer is quiet a large number because 2,3,5 are relatively prime. If we changed the question to the smaller number with perfect roots for powers of 2,3,6 then the answer would be just 64.