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GCF & LCM puzzles, step by step

An ordinary GCF question can be started the moment it is read.

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The question

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. Three facts are given and neither of the two numbers is among them: their greatest common factor is 5, their lowest common multiple is 180, and both are under 50. That last one is not decoration — it is what settles the question at the end.
  2. Their GCF is 5, so both are 5 times something: call them 5×m and 5×n. And m and n can share no factor at all — if they shared one, it would be sitting inside the GCF and 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. The search is now over one small number instead of two unknown ones.
  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, so far all equally good.
  5. Now the last fact does its work: both numbers are under 50, and only 20 and 45 do. That is why the question gave you three facts and not two.

How it works.

  1. 01

    Three facts are given and neither of the two numbers is among them: their greatest common factor is 5, their lowest common multiple is 180, and both are under 50. That last one is not decoration — it is what settles the question at the end.

  2. 02

    Their GCF is 5, so both are 5 times something: call them 5×m and 5×n. And m and n can share no factor at all — if they shared one, it would be sitting inside the GCF and 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. The search is now over one small number instead of two unknown ones.

  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, so far all equally good.

  5. 05

    Now the last fact does its work: both numbers are under 50, and only 20 and 45 do. That is why the question gave you three facts and not two.

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