fnd a number that is a square and a cube and a ninth power?

Question

and it cant be 1

Answer 1

262144

This is 4^9 , 64^3, 512^2...

I basically used guess and check...

Answer 2

Let k^n be this number.
Then, let n be divisible by 2, 3 and 9.
Possible n are multiples of 18.
k can be any number.
Thus, when k = 2, n = 18, we have 2^18.
In general, assuming k and n must be integers, such a number can be expressed as k^18m, where k is a non-negative integer, m is a non-zero integer.
Thus, 5^(180) is the square of 5^90, cube of 5^60 and ninth power of 5^20.

Thus the smallest number is the trivial case, 1.

Answer 3

Basically any number to the 18th power will be a square, cube and ninth power. The reason is that 18 is the lowest common multiple of 2, 3 and 9.

That means:
0^18 --> 0 (aha! nobody else remembered this case)
1^18 --> 1
2^18 --> 262,144
3^18 --> 387,420,489
4^18 --> 68,719,476,736
etc.

Answer 4

That is exactly the same as picking a number that's an 18th power.

2^18, 3^18, 1001^18, (-53)^18 -- any of them will do.

As to WHY it's the same as picking a number that's an 18th power -- in one direction, it's obvious. After all, n^18 is the square of n^9, the cube of n^6, and the 9th power of n^2.

The other direction -- why ANY answer has to be an 18th power -- is pretty easy to see if you know about unique factorization into primes, but pretty mysterious if you don't.

Answer 5

2^18. Its 4 to the 9th power, 64 to the 3rd power and 256 squared.

Answer 6

2^18

Answer 7

Intuitivly, I wd guess ANY number raised to the 'average' of those exponents (2x3x9/3=18)...no?
eg 2^18, which is 262,144


Or do you seek the smallest with this quality?

This is a chemist answering, so answer is subject to review :-))