1. **Problem statement:** Alice wants to buy the same number of apples, oranges, and pears. Apples come in packs of 12, oranges in packs of 7, and pears in packs of 9. We need to find the least number of packs of each fruit so that the total number of each fruit is equal. 2. **Key idea:** The total number of each fruit must be the same. Let this common total be $N$. 3. **Formulate equations:** - Number of apples = $12a$ - Number of oranges = $7b$ - Number of pears = $9c$ We want $12a = 7b = 9c = N$ for some integers $a,b,c$. 4. **Find $N$:** $N$ must be a common multiple of 12, 7, and 9. 5. **Calculate the least common multiple (LCM):** - Prime factors: - $12 = 2^2 \times 3$ - $7 = 7$ - $9 = 3^2$ - LCM takes the highest powers of all primes: - $2^2$ from 12 - $3^2$ from 9 - $7$ from 7 So, $$\text{LCM} = 2^2 \times 3^2 \times 7 = 4 \times 9 \times 7 = 252$$ 6. **Find the number of packs:** - Apples: $a = \frac{N}{12} = \frac{252}{12} = 21$ - Oranges: $b = \frac{N}{7} = \frac{252}{7} = 36$ - Pears: $c = \frac{N}{9} = \frac{252}{9} = 28$ 7. **Answer:** Alice should buy 21 packs of apples, 36 packs of oranges, and 28 packs of pears to have the same number of each fruit. **Final answer:** $$a=21,\quad b=36,\quad c=28$$