<s>[INST] $10000=abc$ 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?
 [/INST] Clearly one of $a,b,c$ is $2^4$
We need to minimize $b+c$ with $bc=5^4$
Now $(b+c)^2-4bc=?\ge0$
So $(b+c)^2\ge4bc=?$ the equality occurs if $b=c$
Fortunately, that condition keeps $b,c$ natural
Else we need to resort to check for possible combinations
$$5^k,5^{4-k},0\le k\le4-k$$
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