Word problem involving the least common multiple of 2 numbers Suppose that hot dogs come in packs of 9 and buns come in packs of 12 . Sam bought the same number of hot dogs as buns. It was the least number of each for which this was possible. Answer the following questions. How many hot dogs did he buy? How many packs of hot dogs did he buy? How many packs of buns did he buy? $$ \square $$ packs Explanation

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The key idea is to find the least common multiple (LCM) of 9 and 12 since Sam bought the same number of hot dogs and buns.

1. Express 9 and 12 in their prime factorizations:
   - $$9 = 3^2$$
   - $$12 = 2^2 \times 3$$

2. The LCM is the product of the highest powers of all prime factors:
   $$
   \text{LCM}(9, 12) = 2^2 \times 3^2 = 4 \times 9 = 36
   $$

3. Sam bought 36 hot dogs (and 36 buns).
   - Number of packs of hot dogs: $$\frac{36}{9} = 4$$
   - Number of packs of buns: $$\frac{36}{12} = 3$$

Thus, Sam bought 36 hot dogs in total, which came in 4 packs of hot dogs and 3 packs of buns.
Sam bought 36 hot dogs in total. He bought 4 packs of hot dogs and 3 packs of buns.

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