Manhattan GMAT Challenge Problem of the Week – 01 Mar 2011

by Manhattan Prep, Mar 1, 2011

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Question

The four members of a jazz quartet all play in different rhythms. The pianist plays in 9/8 time, meaning that his downbeats occur every 9/8 of a measure of time. Meanwhile, the saxophonist plays in 7/4 time, the harpist in 5/8 time, and the drummer in plain old 4/4 time (all with respect to the same measure of time as the pianist). If all four musicians start a song together on the same downbeat, how many measures later will all their downbeats occur simultaneously?

(A) 315

(B) 630

(C) 1,260

(D) 2,520

(E) 5,040

Answer

This problem requires some stripping of the fluff. We have four musicians who play in different rhythms. Consider the pianist: after the initial downbeat, the pianists next downbeat comes at 9/8 of a measure. When would the next one come? At 2 x 9/8, or 18/8. The one after that would be at 3 x 9/8, etc. So were taking multiples of 9/8, to find the downbeat of the pianist.

Likewise, for the other three musicians, their downbeats occur at multiples of their time signatures (7/4, 5/8, and 4/4). To find the first common downbeat after the song has started, we must find the least common multiple of 9/8, 7/4, 5/8, and 4/4.

First, lets make a common denominator for all of these fractions. We should use 8, so we get 9/8, 14/8, 5/8, and 8/8.

Now find the least common multiple of the numerators. The LCM of 9, 14, 5, and 8 is not trivial, because we have a lot of unique prime factors:

9 = 3 x 3

14 = 2 x 7

5 = 5

8 = 2 x 2 x 2

We can drop one 2 (overlap between 14 and 8), but otherwise were stuck. If we try to combine primes to avoid having to multiply two 2-digit numbers together, we get the following. (Hint: save the ugly 7 for last.)

3 x 3 x 2 x 7 x 5 x 2 x 2

= 3 x 3 x 2 x 7 x 10 x 2

= 3 x 3 x 2 x 7 x 20

= 3 x 6 x 7 x 20

= 3 x 7 x 120

= 7 x 360

= 2,520

So the common downbeat will happen at 2,520/8 measures. We must divide 2,520 by 8 now, which yields 315.

The correct answer is A.

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