# Thread: Ten Roots

1. ## Ten Roots

The problem is: Find the smallest integer greater than 1 that has a square root, a cube root, a fourth root, a fifth root, a sixth root, a seventh root, an eighth root, a ninth root, and a tenth root, all perfect.

I have been working at this problem and the only progress I've made is that I know the interger has to end in a 0, 1, 5, or 6. And the integer is BIG, greater than 11^10.

I know that n^2= m^3=x^4 and so on but I don't know if this is useful

2. Originally Posted by math61688
The problem is: Find the smallest integer greater than 1 that has a square root, a cube root, a fourth root, a fifth root, a sixth root, a seventh root, an eighth root, a ninth root, and a tenth root, all perfect.
The number $\displaystyle 2^{2520}$ works.
Do you know why?
Is this the smallest? (use the unique factorization theorem)

3. No I don't know why. That number will work for all?

4. Originally Posted by math61688
No I don't know why. That number will work for all?
Think fractional exponents: $\displaystyle \sqrt[n]{{2^K }} = 2^{\frac{K}{n}}$
Now if $\displaystyle 2^K$ has a perfect sixth root then $\displaystyle 6|K$, 6 divides K.
Do you know/understand the unique factorization theorem?
$\displaystyle 2520$ is the smallest positive integer divisible by each of $\displaystyle ~2,~3,\cdots,~9,~\&~10$.

5. Ok, yes I understand that. It's based on the LCM. Thank you!!