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Zorluk: KolayFactors, Multiples, and Prime Factorization

A red light flashes every 8 seconds, and a blue light flashes every 12 seconds. If they both flash at the same instant, how many seconds will pass before they next flash at the same instant?

  1. A
    4
  2. B
    20
  3. 24Cevap
  4. D
    48
  5. E
    96

Cevap

24 seconds
The correct answer is 24 seconds. To find when both lights will flash at the same time again, we need to find the least common multiple (LCM) of their individual flashing intervals, 8 seconds and 12 seconds. Listing the multiples of each number:
Multiples of 8: 8, 16, 24, 32, 40, 48...
Multiples of 12: 12, 24, 36, 48...
The smallest number that appears in both lists is 24. Thus, the lights will flash together next after 24 seconds.

Adım Adım Çözüm

1
Identify the mathematical concept needed to solve the problem.
The problem requires finding the least common multiple (LCM) of the two flashing intervals, 8 and 12.
The lights will next flash together at the smallest positive integer that is a multiple of both 8 and 12.
2
Find the prime factorization of each number.
8=238 = 2^3 and 12=22×312 = 2^2 \times 3.
Prime factorization allows us to construct the LCM by taking the highest power of each prime factor present in either number.
3
Calculate the LCM from the prime factors.
LCM(8,12)=23×3=8×3=24\text{LCM}(8, 12) = 2^3 \times 3 = 8 \times 3 = 24.
We take the highest power of 2, which is 232^3, and the highest power of 3, which is 313^1.

Anahtar Kavram

Least Common Multiple
Tahmini Süre:45s
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Konular

Factors, Multiples, and Prime Factorization

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