M12-36

Official Solution:

How many positive integers less than 200 are either multiples of 13 or multiples of 12, but not multiples of both?

A. 28
B. 29
C. 30
D. 31
E. 32


The number of multiples of an integer in a range is given by the formula \(\frac{\text{last multiple in the range - first multiple in the range}}{\text{multiple}}+1\). Hence:

• Number of multiples of 13 in the given range \(\frac{(\text{last-first})}{\text{multiple}}+1=\frac{195-13}{13}+1=15\);

• Number of multiples of 12 in the given range \(\frac{(\text{last-first})}{\text{multiple}}+1=\frac{192-12}{12}+1=16\);

• Number of multiples of both 13 and 12 is 1, which is \(13*12=156\).

Note that the number 156 is a multiple of both 13 and 12 and is included in the count of multiples of 13 and the count of multiples of 12. Since we don't want it to be counted at all, we need to subtract it once from the total count of multiples of 13, and once from the total count of multiples of 12.

Therefore, the number of positive integers that are either multiples of 13 or multiples of 12, but not multiples of both, in the given range is \((15-1)+(16-1)=29\).


Answer: B