24 = 2³ x 3
360 = 2³ x 3² x 5

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You can see that both the HCF and the LCM has 2³, so both the original numbers must contain 2³. Because if one of the numbers does not contain 2³, then either the HCF or the LCM will not be satisfied.

Number A: 2³ x unknown expression
Number B: 2³ x unknown expression

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The HCF contains one 3, while the LCM contains only two 3s.

This is an indication that one of the numbers must contain exactly one 3, while the other number must contain exactly two 3s. Otherwise, either the HCF or the LCM will not be satisfied.

Number A: 2³ x 3 x unknown expression
Number B: 2³ x 3² x unknown expression

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Moving on to the 5s. The HCF does not contain any 5 while the LCM contains one 5, so one of the numbers must contain exactly one 5, while the other number must not contain any 5 at all.

Where should the 5 go to?

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Two cases.

Number A: 2³ x 3
Number B: 2³ x 3² x 5

This will be the same two numbers 24 and 360.

The other case is what we would need.

Number A: 2³ x 3 x 5
Number B: 2³ x 3²

So, the two numbers are 120 and 72.