Difference between revisions of "2020 CIME II Problems/Problem 12"
Matboy-123 (talk | contribs) (Created page with "How to do this ??") |
Sweetmango77 (talk | contribs) m |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
| − | + | 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> | ||
| + | |||
| + | 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> | ||
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.
![\[a,b,c = d 20, e 36, f 45\]](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)
