To find the lengths of the tracks Mark and Aki will have when they finish at the same length, we can calculate the least common multiple (LCM) of their track piece lengths.
Mark has pieces that are 4 inches long, and Aki has pieces that are 10 inches long.
- Factors of 4: \( 1, 2, 4 \)
- Factors of 10: \( 1, 2, 5, 10 \)
The LCM can also be calculated using the prime factorization:
- \( 4 = 2^2 \)
- \( 10 = 2^1 \times 5^1 \)
The LCM takes the highest power of each prime:
- For \( 2 \): The highest power is \( 2^2 \).
- For \( 5 \): The highest power is \( 5^1 \).
The LCM is:
\[ LCM = 2^2 \times 5^1 = 4 \times 5 = 20 \]
So the least common multiple of 4 and 10 is 20.
Mark can make tracks of lengths: \( 4, 8, 12, 16, 20, \ldots \) (multiples of 4) Aki can make tracks of lengths: \( 10, 20, 30, \ldots \) (multiples of 10)
They will both have tracks of length 20 inches when they finish at the same length.
Therefore, the answer is 20.
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