How many positive integers less than 200 are there such that

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
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B. 29

for 13: 13...195=13*15 ==> N13=13*15-13/13 +1 = 15
for 12: 12...192=12*16 ==> N13=12*16-12/12 +1 = 16

but there is one integer 13*12. so

N=(15-1)+(16-1)=29

very nice.

i did it the most ungraceful way possible.

200 / 12 = 16.xxxx
200 / 13 = 15.xxx

16 + 15 = 31

12 & 13 have one LCM under 200. didnt even both to calculate it since i know 12*12 =144.

subtract 2 since it shows up in both sets of multiples

31 - 2 = 29

Great Question..
Focus the ording here => NOT BOTH
IF it says EITHER OR => 30
but here its 29
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1)Multiples of 12: (192-12)/12+1=16 2)Multiples of 13: (192-13)/13+1=16 3)Multiple of both 12 and 13 - 2*2*3*13=156 only 1 4)16-1+15-1=29

There is no other way around this question. Use the formula and calculate carefully. The last step is quite easy, look at the unit number only.

Asked: How many positive integers less than 200 are there such that they are multiples of 13 or multiples of 12 but not both?

Number of multiples of 12 = {12,24,....,192} = 16
Number of multiples of 13 = {13,26,,,,,,,195} = 15
Number of multiples of 12*13= 156 = 1

Number of positive integers less than 200 are there such that they are multiples of 13 or multiples of 12 but not both 16+15-2*1 = 29

IMO B

Positive multiples of 12 less than 200 are 16 (as the highest multiple of 12 less than 200 is 192)
Positive multiples of 13 less than 200 are 15 (as the highest multiple of 13 less than 200 is 195)

Now, if a number is a multiple of both 12 and 13, it should be a multiple of 12*13, i.e. 156.

There is only one multiple of 156 less than 200, and that is 156 itself.

Thus, total positive integers less than 200 are there such that
they are multiples of 13 or multiples of 12 but not both = 16 + 15 - 2(since 156 is counted as a multiple in both 12 and 13)
= 29


Thus, the correct option is B.

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