$10000=abc$ with $a,b,c$ natural numbers don't have digit $0$. Smallest possible value of $a+b+c$ is? Let $10000=abc$ with $a,b,c$ natural numbers don't have digit $0$. If $a,b,c$ does not have to be all distinct, then the smallest possible value of $a+b+c$ is?

Attempt:
First write as prime factors: $10000 = 2^{4} 5^{4}$. The possible triples are:
$$ 2, 2^{3}, 5^{4} $$
$$ 2^{2}, 2^{2}, 5^{4} $$
$$ 2^{3}, 2, 5^{4} $$
$$ 2^{4}, 2^{4}, 5^{4} $$
$$ 5, 5^{3}, 2^{4}$$
$$ 5^{2}, 5^{2}, 2^{4}$$
$$ 5^{3}, 5, 2^{4}$$
The smallest sum is $5^{2} + 5^{2} + 2^{4}$. Are there better approaches?

Recognize that the numbers cannot both contain a multiple of $5$ and a multiple of $2$; otherwise they would include a zero. Since $10000=2^4\cdot 5^4$, we know that each of $a,b,c$ is of the form $2^r5^s,$ where $0\leq r,s\leq 4$. Using the above fact we only have three possibilities to consider, though two of them can be ruled out by intuition. $1$) $5^4$ is a term. Then the minimum sum must be greater than $625$. $2$) $5^3$ is a term. Then the minimum sum must be greater than $125$. $3$) $5^2$ is a term. Then another term must be $5^2$ and the remaining term is $2^4$. So the minimum sum is $5^2+5^2+2^4=66.$
