To find the fewest number of slices possible that Amir ended up with the same number of slices of apple pies and peach pies, we need to determine the least common multiple (LCM) of the number of slices per pie for the two types of pies.
Each apple pie has 10 slices, and each peach pie has 15 slices.
To find the LCM of 10 and 15, we can follow these steps:
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Prime Factorization:
- 10 = \(2^1 \times 5^1\)
- 15 = \(3^1 \times 5^1\)
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Identify the highest power of each prime:
- For 2: \(2^1\)
- For 3: \(3^1\)
- For 5: \(5^1\)
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Calculate the LCM: \[ \text{LCM} = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30 \]
Therefore, the fewest number of slices possible that Amir could have, with the same number of apple pie slices and peach pie slices, is:
\[ \boxed{30} \]
This means Amir could have 3 apple pies (3 x 10 = 30 slices) and 2 peach pies (2 x 15 = 30 slices), yielding 30 slices of each type.