What is the shortest string of factors that includes the prime factorizations of both 72 and 120?

Question

Can you find a smaller common mulitple of 72 and 120 and the greatest?

Answer 1

72=2x2x2x3x3
120=2x2x2x3x5
LeastCommon Multiple
=2x2x2x3x3x5
=360
Greatest Common Factor
=2x2x2x3
=24
[12]

Answer 2

72 = 2 x 2 x 2 x 3 x 3 = 2^3 x 3^2
120 = 2 x 2 x 2 x 3 x 5 = 2^3 x 3 x 5
The smallest string that contains both would be :
2^ 3 x 3^ 2 x 5, which is 2 x 2 x 2 x 3 x 3 x 5 which is 360
The greatest common factor is 2^ 3 x 3 which is 2 x 2 x 2 x 3 which is 24.

To find LCM use the HIGHEST power found for each factor in EITHER number. Both had 2^ 3 so use that. The 72 has a 3^ 2, so use that. The 120 has a 5, so use that. So we used 2^ 3 x 3^ 2 x 5 to get the answer of 360.

To find GCF use the LOWEST power found for each factor in BOTH numbers. Both have a 2^ 3, so use that. Both have a 3, so use that. So we used 2^3 x 3 to get the answer of 24.

Answer 3

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Answer 4

72 = 2*2*2*3*3
120 = 2*2*2*3*5

The smallest common multiplier must have at least three 2's, two 3's and one 5.

The biggest common divider cannot have more than three 2's and one 3.