Problem 291

The integers that are multiples of both 5 and 7 are the integers that are multiples of 35.

The smallest multiple of 35 greater than 100 is $3 \times 35 = 105$. (The previous multiple of 35 is $2 \times 35 = 70$.)

Starting at 105 and counting by 35s, we obtain $$105, 140, 175, 210, 245, 280, 315$$ The integers in this list that are between 100 and 300 and are not multiples of 10 (that is, whose units digit is not 0) are 105, 175, 245, of which there are 3.