Problem:
Ralph went to the store and bought pairs of socks for a total of . Some of the socks he bought cost a pair, some of the socks he bought cost a pair, and some of the socks he bought cost a pair. If he bought at least one pair of each type, how many pairs of socks did Ralph buy?
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Solution:
If Ralph buys pairs of socks, then the other pairs of socks would cost at least making the total cost more than . Buying fewer than pairs of socks would make Ralph's cost even higher. If he bought pairs of socks, then the other pairs would cost less than making the total cost less than . Buying more than pairs of socks would make his total cost even lower. So Ralph bought pairs of socks, pairs of socks, and pairs of socks.
Let and be the number of pairs of and socks, respectively. Then and . Subtracting the first equation from the second gives . Since is a factor of both and must also be a factor of . Since , it follows that , and .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions