[ [Divisibility of numbers. General properties]

Authors: Binkov A.d,, Raskina I.v.

Several whole heads of cheese were stored in the warehouse. At night, rats came and ate 10 heads, and all of them ate equally. Several rats got sick from overeating. The remaining seven rats the next night finished off the remaining cheese, but each rat was able to eat only half as much cheese as the night before. How much cheese was originally in the warehouse?

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Let there be $k$ rats in total ($k>7$), then each rat ate $10 / k$ pieces of cheese on the first night. On the second night, each rat ate half as much, i.e., $5 / k$ pieces. Seven rats thus ate $35 / k$ pieces. This is an integer.

The only divisor of the number 35 that exceeds 7 is the number 35 itself. Therefore,

$35 / k=1$, and there were $10+1=11$ pieces of cheese in the warehouse before the rat invasion.

## Answer

11 pieces of cheese