2001 AMC 10 Problems/Problem 8

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Problem

Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?

$\textbf{(A) }42\qquad\textbf{(B) }84\qquad\textbf{(C) }126\qquad\textbf{(D) }178\qquad\textbf{(E) }252$

Solution

We need to find the least common multiple of the four numbers given. That is, the next time they will be together. First, find the least common multiple of $3$ and $4$.

$3 \times 4 = 12$.

Find the least common multiple of $12$ and $6$. Since $12$ is a multiple of $6$, the least common multiple is $12$. Lastly, the least common multiple of $12$ and $7$ is $12 \times\ 7 = \boxed{\textbf{(B) }84}$.

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