Multiples Math Example 4

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Example 4

medium
A light flashes every 44 seconds and a buzzer sounds every 66 seconds. They start together. When will they next coincide? How many times in one minute?

Solution

  1. 1
    They coincide at common multiples of 44 and 66. lcm(4,6)=12\text{lcm}(4,6) = 12 seconds.
  2. 2
    They next coincide after 1212 seconds.
  3. 3
    In 6060 seconds: 60÷12=560 \div 12 = 5 times (at t=12,24,36,48,60t = 12, 24, 36, 48, 60 s). Including t=0t = 0: 66 coincidences, but the question asks for how many times after the start: 55.

Answer

Next coincidence after 1212 seconds; they coincide 55 times in one minute (not counting t=0t=0).
When two periodic events must occur simultaneously, their coincidences happen at multiples of the LCM of their periods. This is a classic application of least common multiples in real-world timing problems.

About Multiples

Numbers obtained by multiplying a given number by positive integers: the skip-counting sequence n,2n,3n,4n,…n, 2n, 3n, 4n, \ldots

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More Multiples Examples

Example 1 easy

List the first [formula] multiples of [formula], and find the [formula]th multiple of [formula].

Example 2 medium

Find all multiples of [formula] between [formula] and [formula] (inclusive). How many are there?

Example 3 easy

Is [formula] a multiple of [formula]? Of [formula]? Show your reasoning.

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