2020 CIME II Problems/Problem 12: Difference between revisions
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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. | |||
<cmath>a,b,c = d 20, e 36, f 45</cmath> | |||
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. | |||
<cmath>a,b,c = 40,45,108</cmath> | |||
<cmath>a+b+c = 193</cmath> | |||
Revision as of 08:20, 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.
![\[a,b,c = d 20, e 36, f 45\]](http://latex.artofproblemsolving.com/9/6/a/96a447d3a1d1934ce75df37d304441f0437d857d.png)
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.
![\[a,b,c = 40,45,108\]](http://latex.artofproblemsolving.com/c/3/9/c3938226c109ba1830dceefb12a97ae31586bd69.png)
![\[a+b+c = 193\]](http://latex.artofproblemsolving.com/b/0/9/b09cde7e3d6a41372e85d09ee3befce4df3fedfc.png)
