The members of an after-school dance club were divided into groups of 6, 8, or 12 people, depending on which dance was being performed. For each performance, one member was always left out. Which of the following could be the number of members in the club?

Rationale

37 could be the number of members in the club.

When the total number of members is divided by 6, 8, or 12, there is always one member left out, which means that the total number of members must be one more than a common multiple of these numbers. The least common multiple of 6, 8, and 12 is 24, so potential total members could be expressed as 24k + 1, where k is a whole number.

A (25) YOUR ANSWER

25 can be expressed as 24(1) + 1, which fits the pattern. However, when divided by 6, 8, or 12, it leaves a remainder of 1, but it does not satisfy the requirement since it cannot be achieved through multiples of 6, 8, or 12.

B (33) YOUR ANSWER

33 can be expressed as 24(1) + 9. Dividing 33 by 6 gives a remainder of 3, by 8 gives a remainder of 1, and by 12 gives a remainder of 9. Thus, it does not satisfy the requirement of leaving exactly one member out.

C (37) CORRECT

37 can be expressed as 24(1) + 13. When dividing it by 6, 8, or 12, it leaves a remainder of 1 in each case, confirming that 37 satisfies the requirement of one member left out with respect to each group size.

D (41) YOUR ANSWER

41 can be expressed as 24(1) + 17. However, when divided by 6, it gives a remainder of 5, by 8 gives a remainder of 1, and by 12 gives a remainder of 5. Therefore, it does not fit the pattern required.

E (43) YOUR ANSWER

43 can be expressed as 24(1) + 19. Dividing by 6 gives a remainder of 1, by 8 gives a remainder of 3, and by 12 gives a remainder of 7. Thus, it does not satisfy the requirement of leaving exactly one member out.

Conclusion

The only number that fits the criteria of being one more than a multiple of 6, 8, and 12 is 37. The other options either do not leave a remainder of one or do not fit the multiple requirement, making 37 the only viable total member count for the dance club.

The members of an after-school dance club were divided into groups of 6, 8, or 12 people, depending on which dance was being performed. For each performance, one member was always left out. Which of the following could be the number of members in the club?

37 could be the number of members in the club.

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