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Julian owns a business and assigns two of his employees their shifts. He assigns the first employee shifts in blocks of $3$ hours, and he assigns the second worker shifts in blocks of $12$ hours. If both workers must receive the same number of hours, what is the minimum number of hours each must be assigned? $\newline$ $\_\_\_\_\_\_\_$ hours

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GCF and LCM: word problems

Full solution

Q. Julian owns a business and assigns two of his employees their shifts. He assigns the first employee shifts in blocks of

3

hours, and he assigns the second worker shifts in blocks of

12

hours. If both workers must receive the same number of hours, what is the minimum number of hours each must be assigned?

\newline

\_\_\_\_\_\_\_

hours

Identify Concept: Identify the mathematical concept needed to solve the problem. Since both employees must receive the same number of hours, and their shifts are in blocks of different lengths ( $3$ hours and $12$ hours), we need to find the Least Common Multiple (LCM) of $3$ and $12$ .
Calculate LCM: Calculate the LCM of $3$ and $12$ . List the multiples of each: $\newline$ Multiples of $3$ : $3, 6, 9, 12, 15, 18, 21, 24\ldots$ $\newline$ Multiples of $12$ : $12, 24, 36, 48\ldots$ $\newline$ The smallest common multiple is $12$ .

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