I'm studying, and got this problem in the book,
How many positive integers between 50 and 100 are divisible by 7, and what are they?
The same for number divisible by both 7 and 11.
I'm reviewing Counting Basics, and this looks like counting subsets on a finite set, but I'm not sure how to solve this (I've seen combinations, like strings, but not this). Can someone help me get started?
If you'd like, the answers are: 7 for the first one (56, 63, 70, 77, 84, 91, 98), and 1 for the second (77). But again, I'm interested in the process itself.
Best Answer
Your best bet in a situation like this, where there are relatively few numbers between $50$ and $100$ which are divisible by $7$, is to just list them, and count the number of entries on the list. (Of course, you'd like a more efficient means if the interval over which multiples of a given number span is very very large!) Note: When computing the number of integers between $50$ and $100$ that are divisible by both $7$ and $11$, there is only one such number: $7 \times 11 = 77$.
If, however, the question asked how many such numbers are divisible by $7$ OR $11$, then you'd need to list/count
(1) those divisible by $7$,
(2) those divisible by $11$;
(3) those divisible by both $7$ and $11$: any multiple of $7 \times 11$ in the given range;
Then add the number of entries on list (1) to those on (2), and then subtract the number of entries on (3) from that sum. (Otherwise, without subtracting, you'd be counting, e.g. 77, twice, since it would appear on both (1) and (2).)
In general, if the lower bound of the interval under consideration is $L$, and the upper bound of the interval under consideration is $U$, then the smallest multiple of a positive integer $n$ that is larger than or equal to $L$ is $L/n$, and if the result is not an integer, round up to the next integer (say $k_{min}$). On the other hand, the largest multiple of $n$ that is less than or equal to $U$ is $U/n$, but if the result is not an integer, round down (drop off the remainder), say $k_{max}$. Then, count all multiples of $n$ such that $L \leq nk_{min} < nk_i < nk_{max} \leq U$, where $nk_i$ are all the multiples of $n$ between the least and greatest integers in the given interval [$L, U$].