A large purse is full of coins. If you count them by 13's, 23's, or 31's, there will be one left over. If you count them by 73's there will be none left over. How many coins are there in the purse?
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Suppose there are x coins in the purse.
From the condition, we have:
(1) x mod 13=1, ...
Mods are used to solve a word problem.