How can I find the possible values that $\gcd(a^3,b^4)$ can take, if $\gcd(a,b)=10$? I don´t know how to find them, any ideas would really help. 
 A: Hint: Since $10$ divides $a$ and $b$, we can write $\;a = 10k,\; b = 10j\;$ where $k, j$ are coprime integers (else $\gcd(a, b) > 10$). So $$\;a^3 = (10k)^3 = 1000k^3;\quad \;b^4 = (10j)^4 = 10000j^4.$$ 
Now consider $$\;\gcd(a^3, b^4) = \gcd(1000k^3, 10000j^4).$$
A: First idea: what's the biggest number you can think of that absolutely has to divide both $a^3$ and $b^4$, if $10$ divides both $a$ and $b$?
Second idea: if you changed the problem to $\gcd(a^4,b^4)$, would it be easier? What would the answer be? It might not be the same answer as for $\gcd(a^3,b^4)$, but it's certainly a multiple of $\gcd(a^3,b^4)$, since $a^3\mid a^4$.
Together, the two ideas should give you both a lower bound and an upper bound for $\gcd(a^3,b^4)$. Then you can start looking for examples that give you the various possibilities in that range (or maybe you'll find a reason why certain such possibilities never occur).
A: Hint $\rm\ 10^3\!=(a,b)^3\mid (a^3,b^4)\mid (a^4,b^4)=10^4\:$ so, by unique factorization, the only possible values are $\rm\,10^3\{1,2,5,10\},\:$ which are all realized for the values $\rm\ a,b\, =\, 10,10;\ \ 20,10;\ \ 50,10;\ \ 100,10.$
A: Let's think about the divisors :
Since ($a$,$b$) is equal to 10, and you have ($a^3$,$b^{3}$) as a subset, then you have $10^1$,$10^2$,$10^3$, and since you also have and extra b, you have an extra 10, you can can go up to $10^4$.
But also consider, the factors of 10: $2 \cdot 5$, so we also have :
$2^1$,$2^2$,$2^3$,$2^4$,
and
$5^1$,$5^2$,$5^3$,$5^4$,
And all their combinations. However the 10 will be the greatest of these, so you will get one of :
($10^4,10^3,10^3 \cdot 2,10^3 \cdot 5$)
