Difference between revisions of "2019 AIME I Problems/Problem 7"
Expilncalc (talk | contribs) (→Solution: Added solution.) |
|||
| Line 1: | Line 1: | ||
| − | |||
| − | |||
==Problem 7== | ==Problem 7== | ||
There are positive integers <math>x</math> and <math>y</math> that satisfy the system of equations <cmath>\log_{10} x + 2 \log_{10} (\gcd(x,y)) = 60</cmath><cmath>\log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) = 570.</cmath> Let <math>m</math> be the number of (not necessarily distinct) prime factors in the prime factorization of <math>x</math>, and let <math>n</math> be the number of (not necessarily distinct) prime factors in the prime factorization of <math>y</math>. Find <math>3m+2n</math>. | There are positive integers <math>x</math> and <math>y</math> that satisfy the system of equations <cmath>\log_{10} x + 2 \log_{10} (\gcd(x,y)) = 60</cmath><cmath>\log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) = 570.</cmath> Let <math>m</math> be the number of (not necessarily distinct) prime factors in the prime factorization of <math>x</math>, and let <math>n</math> be the number of (not necessarily distinct) prime factors in the prime factorization of <math>y</math>. Find <math>3m+2n</math>. | ||
Revision as of 21:41, 14 March 2019
Problem 7
There are positive integers 
and 
that satisfy the system of equations ![\[\log_{10} x + 2 \log_{10} (\gcd(x,y)) = 60\]](http://latex.artofproblemsolving.com/1/6/f/16f085cc382249acf20a8b88a9a35f00cbac2fb3.png)
![\[\log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) = 570.\]](http://latex.artofproblemsolving.com/e/1/e/e1eff7ffc812984b5b58c17079adcc6677c3497b.png)
Let 
be the number of (not necessarily distinct) prime factors in the prime factorization of , and let 
be the number of (not necessarily distinct) prime factors in the prime factorization of . Find 
.
Solution
One immediately sees that 
and 
, so the answer is 
because 
and similarly for 
.
See Also
| 2019 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
