# Difference between revisions of "2020 CIME II Problems/Problem 12"

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− | Note that the <math>lcm(gcd(a,b),c)</math> and its iterations are all divisible by 180. This implies that 2 of <math>a,b,c</math> are divisible by 4, 2 are divisible by 9 and 2 are divisible by 5. | + | Note that the <math>\text{lcm}(\text{gcd}(a,b),c)</math> and its iterations are all divisible by 180. This implies that 2 of <math>a,b,c</math> are divisible by 4, 2 are divisible by 9 and 2 are divisible by 5. |

<cmath>a,b,c = d 20, e 36, f 45</cmath> | <cmath>a,b,c = d 20, e 36, f 45</cmath> | ||

## Latest revision as of 10:27, 19 January 2021

Note that the and its iterations are all divisible by 180. This implies that 2 of are divisible by 4, 2 are divisible by 9 and 2 are divisible by 5.

Next we note that iterations are 180,,. This implies that or must have an additional factor of 2 and or must have an additional factor of 3. The sum is minimized if d=2 and e=3 and f=1.