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

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How to do this ??
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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.
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<cmath>a,b,c = d 20, e 36, f 45</cmath>
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Next we note that iterations are 180,<math>2\times 180</math>,<math>3 \times 180</math>. This implies that <math>d</math> or <math>e</math> must have an additional factor of 2 and <math>e</math> or <math>f</math> must have an additional factor of 3. The sum is minimized if d=2 and e=3 and f=1.
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<cmath>a,b,c = 40,45,108</cmath>
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<cmath>a+b+c = 193</cmath>

Latest revision as of 10:27, 19 January 2021

Note that the $\text{lcm}(\text{gcd}(a,b),c)$ and its iterations are all divisible by 180. This implies that 2 of $a,b,c$ are divisible by 4, 2 are divisible by 9 and 2 are divisible by 5. \[a,b,c = d 20, e 36, f 45\]

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

\[a,b,c = 40,45,108\] \[a+b+c = 193\]

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