kevincan wrote:
How many positive integers less than 1000 are multiples of 5 but NOT of 4 or 7?
(A) 114
(B) 121
(C) 122
(D) 129
(E) 136
Let’s first determine the number of multiples of 5 from 5 to 995, inclusive:
(995 - 5)/5 + 1 = 199
If a number is a multiple of 5, but not a multiple of 4, then it’s not a multiple of 20. So let’s determine the number of multiples of 20 less than 1000:
(980 - 20)/20 + 1 = 49
If a number is a multiple of 5, but not a multiple of 7, then it’s not a multiple of 35. So let’s determine the number of multiples of 35 less than 1000:
(980 - 35)/35 + 1 = 28
Thus, we need to subtract 49 and 28 from 199:
199 - 49 - 28 = 122
However, we “over” subtracted from 199; for example, we subtracted 980 twice, as it is a multiple of 20 and also a multiple of 35. In other words, we “over” subtracted the number of numbers that are multiples of 5, 4 and 7, i.e., a multiple of the LCM of 5, 4, 7, which is 140. So we have to determine the number of multiples of 140 less than 100 and add that number back to 122:
(980 - 140)/140 + 1 = 7
Thus, the number of positive integers less than 1000 which are a multiple of 5 but not 4 or 7 is 122 + 7 = 129.
Answer: D
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