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GCF & LCM puzzles

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The question

Two numbers have a greatest common factor of 5 and a least common multiple of 180.

Both numbers are less than 50. What are the two numbers?

Both numbers are the GCF times something

GCF:5· LCM:180
5×mand5×n
180÷5=36
1×36→5and180
4×9→20and45
5and180
20and45
  1. We know three things: the greatest common factor is 5, the lowest common multiple is 180, and both numbers are under 50.
  2. Their GCF is 5, so both are 5 times something: call them 5×m and 5×n. m and n share no factor — otherwise the GCF would be bigger than 5.
  3. For two numbers built that way the LCM is 5 × m × n. So 5 × m × n = 180, which means m × n = 180 ÷ 5 = 36.
  4. List the ways to make 36 from two numbers sharing no factor, and multiply each back up by 5: that gives 5 and 180, 20 and 45. Every one has GCF 5 and LCM 180.
  5. The last fact decides it: both numbers are under 50, and only 20 and 45 do.

How it works.

  1. 01

    We know three things: the greatest common factor is 5, the lowest common multiple is 180, and both numbers are under 50.

  2. 02

    Their GCF is 5, so both are 5 times something: call them 5×m and 5×n. m and n share no factor — otherwise the GCF would be bigger than 5.

  3. 03

    For two numbers built that way the LCM is 5 × m × n. So 5 × m × n = 180, which means m × n = 180 ÷ 5 = 36.

  4. 04

    List the ways to make 36 from two numbers sharing no factor, and multiply each back up by 5: that gives 5 and 180, 20 and 45. Every one has GCF 5 and LCM 180.

  5. 05

    The last fact decides it: both numbers are under 50, and only 20 and 45 do.

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