# 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 = a

^{2}= b^{3}= c^{5}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.*:
6

119

14

1,007

^{5}= 7,776119

^{5}= 23,863,536,59914

^{5}= 537,8241,007

^{5}= 1,035,493,442,021,807Advertisement:

## Solution

We could brute force this, but we donâ€™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 = a

^{2}= b^{3}= c^{5}= x^{z}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 = x

^{30}Because of the way powers work, we can split this common multiple into the components we need:

N = (x

^{2})^{15}= (x^{3})^{10}= (x^{5})^{6}Which of course is equivalent to:

N = (x

^{15})^{2}= (x^{10})^{3}= (x^{6})^{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 = (2

1,073,741,824 = 32,768

^{15})^{2}= (2^{10})^{3}= (2^{6})^{5}1,073,741,824 = 32,768

^{2}= 1,024^{3}= 64^{5 }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.