How I would have to solve it, what do you think?
I need to find the smallest natural number X [ ] with these features:
1) is a square of whole number
2) is a cube of whole nuber
3) is a fifth degree of whole number
In the natural numbers, N, find the lowest such that half of it is a perfect square, one-third is a perfect cube, and one fifth is a perfect fifth power
These features have to be true all at the same time.
Then 2 must divide x, 3 must divide x, and 5 must divide x.
Therefore x is of the form 2.3.5.(product of prime numbers).
After dividing by 2 all powers of the primes in the factorization must be even.
So x must be of the form 2^odd . 3^even . 5^even . (other primes)^even.
After dividing by 3 the powers must be multiples of 3.
So x must be of the form 2^(3k) . 3^(3l+1) . 5^(3m) . (other primes)^(multiple of 3).
After dividing by 5 the powers must be multiples of 5.
So x must be of the form 2^(5k') . 3^(5l') . 5^(5m'+1) . (other primes)^(multiple of 5).
To find the smallest number, we need the lowest powers.
And the numbers 2, 3, and 5 need a power of at least 1.
The smallest number that satisfies that is:
x = 2^15 . 3^10 . 5^6