1968 IMO Problems/Problem 2
Find all natural numbers such that the product of their digits (in decimal notation) is equal to .
Let the decimal expansion of be , where are base-10 digits. We then have that . However, the product of the digits of is , with equality only when is a one-digit integer. Therefore the product of the digits of is always at most , with equality only when is a base-10 digit. This implies that , so . Every natural number from 1 to 12 satisfies this inequality, so we only need to check these possibilities. It is easy to rule out 1 through 11, since for those values. However, , which is the product of the digits of 12. Therefore is the only natural number with the desired properties.
Now note that, if is a prime such that then .
But, which means don't divivde
It is easy to see that has one solution and that is ( Prove it by contradiction)
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