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How many positive integers between 50 and 100

a) are divisible by 7? Which integers are these?

This question is in the basic counting section of my textbook and I'm just studying for finals now. For this question, I was wondering if there is a less primitive way of finding this answer. I did:

56, 63, 70, 77, 84, 91, 98; These answers I found by just going on with my 7's multiplications table.

The reason I'm wondering this is in case I get asked the same thing but with a much larger number such as 1000.

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If it asks you to list them as this question does then you'll have to do it the hard way anyway.... – Simon Hayward Dec 8 '12 at 21:55
up vote 8 down vote accepted

$$50 = 7 \times 7 +1$$ $$100 = 14 \times 7 +2$$ and the answer is $14-7=7$.

This is effectively what you have done, counting $8 \times 7$, $9 \times 7$, $10 \times 7$, $11 \times 7$, $12 \times 7$, $13 \times 7$, and $14 \times 7$. You will need to be more careful if either of the remainders is $0$.

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If I understand this correctly, you are using the min range and max range and given the integer value, use what product to equal those ranges, then subtracting those products to get the number between the min and max range? – Ben Sewards Dec 9 '12 at 2:29
@Ben: Yes: since $100 = 14 \times 7 +2$ there are $14$ positive multiples of $7$ below $100$, but $7$ of these are below $50 = 7 \times 7 +1$, implying there are $14-7=7$ between $50$ and $100$. – Henry Dec 9 '12 at 8:53
Thank you for the clarification – Ben Sewards Dec 9 '12 at 18:11

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