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CEMC Gauss (Grade 8), 2014

Number theory Difficulty 1.2 Junior Find the answer Canada

How many prime numbers are there between 10 and 30?

Pick one

Solution

The largest two-digit number that is a multiple of 7 is $7\times14=98$ .

Thus, there are 14 positive multiples of 7 that are less than 100.

However, this includes $7\times1=7$ which is not a two-digit number.

Therefore, there are $14-1=13$ positive two-digit numbers which are divisible by 7.
(Note that these 13 numbers are $14,21,28,35,42,49,56,63,70,77,84,91,$ and 98.)

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