# SAT multiples problem

• May 21st 2012, 06:36 PM
m58
SAT multiples problem
If $S$ is the set of positive integers that are multiples of $7$, and if $T$ is the set of positive integers that are multiples of $13$, how many integers are in the intersection of $S$ and $T$?

I said there is 1 integer in the intersection, since only 91 is a common multiple of 7 and 13. However, according to the answer,

Quote:

This is the set of all positive integers that are multiples of $7*13 = 91$. There are an infinite number of positive integers that are multiples of $91$, so there are more than thirteen integers in the intersection of $S$ and $T$.
91's multiples include 186, 278, etc, so I'm guessing the infinity is referring to that. However, I thought the problem only wanted an integer that was only multiple of 7 and 13?
• May 22nd 2012, 10:07 AM
HallsofIvy
Re: SAT multiples problem
Quote:

Originally Posted by m58
If $S$ is the set of positive integers that are multiples of $7$, and if $T$ is the set of positive integers that are multiples of $13$, how many integers are in the intersection of $S$ and $T$?

I said there is 1 integer in the intersection, since only 91 is a common multiple of 7 and 13. However, according to the answer,

91's multiples include 186, 278, etc, so I'm guessing the infinity is referring to that. However, I thought the problem only wanted an integer that was only multiple of 7 and 13?

You have dropped and "s"! The problem said "multiples" of 7 and 13, not just the product of 7 and 13. The set of "positive multiples of 7" are 7, 14, 21, 28, etc. and the set of "positive multiples of 13" are 13, 26, 39, etc.
• May 23rd 2012, 07:10 AM
m58
Re: SAT multiples problem
Quote:

Originally Posted by HallsofIvy
You have dropped and "s"! The problem said "multiples" of 7 and 13, not just the product of 7 and 13. The set of "positive multiples of 7" are 7, 14, 21, 28, etc. and the set of "positive multiples of 13" are 13, 26, 39, etc.

So the multiples of 7 and 13 that are x≥91 are included in the intersection, meaning there are infinitely many of them.