HMMT-Feb 2010 guts, Problem 6

How many different numbers are obtainable from five $5 \mathrm{~s}$ by first concatenating some of the $5 \mathrm{~s}$, then multiplying them together? For example, we could do $5 \cdot 55 \cdot 55,555 \cdot 55$, or 55555 , but not $5 \cdot 5$ or 2525 .

Solution: If we do 55555 , then we're done. Note that $5,55,555$, and 5555 all have completely distinguishable prime factorizations. This means that if we are given a product of them, we can obtain the individual terms. The number of 5555's is the exponent of 101, the number of 555's is the exponent of 37, the number of 55's is the exponent of 11 minus the exponent of 101, and the number of 5's is just whatever we need to get the proper exponent of 5 . Then the answer is the number of ways we can split the five 5's into groups of at least one. This is the number of unordered partitions of 5 , which is 7 . $\fbox{7}$.

Answer: 7