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I can't see any relationship between the products of the positive factors of powers of 10. The answer given here is 50, by the way, but I simply don't know where to start.

Nameless King
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    Hint: Almost all factors can be paired up $(a,b)$ such that $ab = 10^4$. So this really boils down to: which one(s) can't? And how many pairs of factors can we make? – platty Dec 12 '18 at 00:04
  • $10^4=2^4\cdot 5^4$, now all combinations of $2^k\cdot 5^j$ for $j,k\leq 4$ are your options here. If you use brute force (basically some technical combinatorics) you can now get to the result you want using $\log$'s properties (and use some symmetries to make it easier to calculate $\log_{10}$ of the things you get). Might be a more elegant way but this should also work. – NL1992 Dec 12 '18 at 00:08

2 Answers2

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As @NL1992 pointed out, the key is write $10^4 = 2^4 \times 5^4$.

Consider the factors which have no contribution from the "2". These are $5^0, 5^1, 5^2, 5^3$, and $5^4$. Their product is $5^{10}$.

Now, consider the factors which exactly one "2". These are $2 \times 5^0$, $2 \times 5^1$, $2 \times 5^2$, $2 \times 5^3$, and $2 \times 5^4$. Their product is $2^5 \times 5^{10}$.

Generalizing, you can see that if you multiply all factors which have a $2^k$ term, you get $2^{5k} \times 5^{10}$.

So the final answer is $n = \prod_{k=0}^4 2^{5k}5^{10} = 2^{50} 5^{50}$, implying $\log_{10}(n) = 50$.

Godfather
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Hint:

We can see, either through investigation or just trying a few examples if you want, that, if a number $n$ has $d(n)$ divisors (sometimes denoted $\sigma_0(n)$ instead), then we can give the product of $n$'s divisors by

$$\prod_{d|n} d = n^{d(n) / 2}$$

This is discussed in more length at Product of Divisors of some $n$ proof

Personally I've seen this occasionally denoted $\pi(n)$ and used such a notation myself when I was looking into the product of a number's divisors. Either way, not important. But with that in mind, you are seeking

$$\log_{10} \left( \pi(10^4) \right) \;\;\; \text{or equivalently} \;\;\; \log_{10} \left( \prod_{d|10^4} d \right)$$

With the previous fact in mind, this problem is essentially reduced to simply finding the number of divisors of $10^4$ after using a few basic properties of logarithms and exponents.

You should be able to easily continue from here.

PrincessEev
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