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The two numbers 18 and 27.

Factorising the lcm and gcf gives:

54 = 2 × 3³

9 = 3²

There must be 3² in both numbers, but there must be a 3³ and a 2 which must appear in the prime factorisations of the two numbers. Putting a 2 with one 3² and a 3 with the other 3² leads to the two numbers:

2 × 3² = 18

3 × 3³ = 3³ = 27

as the two smallest numbers with gcf=9 and lcm=54. The other distribution of both 2 and 3 with one 3² results in the two number 3² = 9 and 2 × 3³ = 54, the latter of which is larger than both the 18 and 27 previously found.

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9y ago

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