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Zane notices an identical number of two types of insects in his neighborhood: butterflies and fireflies. He always seems to observe butterflies in groups of $5$ and fireflies in groups of $10$ . What is the smallest number of butterflies that he could have seen? $\newline$ $\_\_\_\_\_\_\_\_\_$ butterflies

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GCF and LCM: word problems

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Q. Zane notices an identical number of two types of insects in his neighborhood: butterflies and fireflies. He always seems to observe butterflies in groups of

5

and fireflies in groups of

10

. What is the smallest number of butterflies that he could have seen?

\newline

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

butterflies

Identify pattern: Identify the pattern of observation for butterflies and fireflies. Butterflies are seen in groups of $5$ , and fireflies in groups of $10$ . We need to find the smallest number of butterflies that fits both patterns.
Determine LCM: Determine the Least Common Multiple (LCM) of the group sizes for butterflies ( $5$ ) and fireflies ( $10$ ) to find the smallest group size that fits both observations. Calculation: LCM of $5$ and $10$ .
Calculate LCM: Since $10$ is a multiple of $5$ , the LCM of $5$ and $10$ is $10$ . This means the smallest number of butterflies Zane could have seen that also fits the pattern for fireflies is $10$ .

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