Discussion
How many integers are there between 1 and 100 which have 5 as a digit but are not divisible by 7?
Answer: A
There are 17 such integers.
First list the numbers between 1 and 100 with a 5 as a digit: 5, 15, 25, 35, 45, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 65, 75, 85 and 95, which is 19 numbers (55 counted once).
Of these, the multiples of 7 are 35 and 56. Removing them leaves \(19 - 2 = 17\).
18 comes from removing only one multiple of 7 (usually 35, missing 56), and 14 or 15 from undercounting the fifties. Listing is safer than a formula here.
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