Multiples Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Find all multiples of 66 between 5050 and 100100 (inclusive). How many are there?

Solution

  1. 1
    First multiple of 66 at or above 5050: ⌈50/6⌉=9\lceil 50/6 \rceil = 9, so 6×9=546 \times 9 = 54.
  2. 2
    Last multiple of 66 at or below 100100: ⌊100/6⌋=16\lfloor 100/6 \rfloor = 16, so 6×16=966 \times 16 = 96.
  3. 3
    Multiples: 54,60,66,72,78,84,90,9654, 60, 66, 72, 78, 84, 90, 96.
  4. 4
    Count: 16−9+1=816 - 9 + 1 = 8.

Answer

54,60,66,72,78,84,90,9654, 60, 66, 72, 78, 84, 90, 96 — eight multiples of 66 in [50,100][50, 100].
To count multiples of nn in a range [a,b][a, b], divide to find the first and last indices, then subtract and add 11. The formula is ⌊b/n⌋−⌈a/n⌉+1\lfloor b/n \rfloor - \lceil a/n \rceil + 1 when the range includes multiples.

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 3 easy

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

Example 4 medium

A light flashes every [formula] seconds and a buzzer sounds every [formula] seconds. They start toge

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