João has more than 30 and less than 100 chocolates. If he arranges the chocolates in rows of 7, one will be left over. If he arranges them in rows of 10, two will be left over. How many chocolates does he have?

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Solution

In the first organization, with $x$ being the number of rows, the number of chocolates João has is of the form $7 x+1$. In the second organization, with $y$ being the number of rows, the number of chocolates João will have is $10 y+2$. That is, the number of chocolates João has leaves a remainder of 1 when divided by 7 and a remainder of 2 when divided by 10. In the range between 30 and 100, there are 7 numbers that leave a remainder of 2 when divided by 10: 32, 42, 52, 62, 72, 82, and 92. Among these numbers, only one leaves a remainder of 1 when divided by 7: 92. Therefore, the number of chocolates João has is 92.