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Zorluk: OrtaFactors, Multiples, and Primes

In a piece of heavy machinery, two interlocking gears are rotating together. The drive gear has 2828 teeth, and the driven gear has 4242 teeth. At a certain moment, a marked tooth on the drive gear aligns exactly with a marked tooth on the driven gear. How many teeth must pass the alignment point before these two specific marked teeth align with each other again?

  1. 8484Cevap
  2. B
    1414
  3. C
    11761176
  4. D
    7070

Cevap

The gears will align again after 8484 teeth pass the alignment point.
To find the moment when two rotating cycles synchronize, the Least Common Multiple (LCM) of their cycle lengths (the number of teeth) must be determined. By finding the prime factorizations (28=22×728 = 2^2 \times 7 and 42=2×3×742 = 2 \times 3 \times 7) and multiplying the highest powers of all unique prime factors present (22×3×72^2 \times 3 \times 7), the LCM is calculated as 8484. Thus, exactly 8484 teeth must pass for the marked teeth to realign.

Adım Adım Çözüm

1
Identify the mathematical concept required to solve the synchronization problem.
The problem asks for the next time repeating cycles coincide, which requires finding the Least Common Multiple (LCM) of 2828 and 4242.
The LCM determines the smallest positive integer that is a multiple of both numbers, representing the first point where both gear cycles overlap.
2
Find the prime factorization of the number of teeth on each gear.
For the drive gear: 28=22×728 = 2^2 \times 7. For the driven gear: 42=2×3×742 = 2 \times 3 \times 7.
Prime factorization breaks down each number into its core primes, allowing for an accurate calculation of the LCM.
3
Calculate the LCM using the identified prime factors.
LCM = 22×3×7=4×3×7=842^2 \times 3 \times 7 = 4 \times 3 \times 7 = 84.
The LCM is calculated by taking the highest power of each prime factor that appears in any of the original factorizations.

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Factors, Multiples, and Primes
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Factors, Multiples, and Primes

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