Bunuel wrote:
In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, those whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects?
A. 19
B. 21
C. 26
D. 41
E. 57
Our goal is to find the number of students who do not opt for any of the three subjects. We first can find the number of students who do not opt for physics (i.e., eliminate the number of students who opt for it). Then, from those students, we eliminate those who opt for chemistry. Finally, from those who are left (after eliminating physics and chemistry), we eliminate those who opt for math. Thus, the students who are left are those who do not opt for any of the three subjects.
Since all even-numbered students (60 students) opt for physics, we know the odd-numbered students (the other 60 students) do not opt for physics. That is, the students numbered 1, 3, 5, …, 119 do not opt for physics. From these students, we see that the odd multiples of 5 (5, 15, 25, …, 115) opt for chemistry, and thus we have to eliminate them. The number of these students is:
(115 - 5)/10 + 1 = 12
Thus, we have 60 - 12 = 48 students left who do not opt for either physics or chemistry (or both). From these students, we need to eliminate those who are multiples of 7, since they opt for math. The numbers must be odd multiples of 7, namely, 7, 21, 35, 49, 63, 77, 91, 105, and 119. There are 9 such numbers. However, we see that we’ve already counted 35 and 105 since they are odd multiples of 5. Thus, there are 7 odd multiples of 7 that are not odd multiples of 5, and we have to exclude them. Thus, we have 48 - 7 = 41 students left and these students do not opt for any of the three subjects.
Answer: D
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