Difference between revisions of "2018 AMC 10B Problems/Problem 23"
Ironicninja (talk | contribs) (Original had 21*8 = 189 which is not true. Also, fixed the latex formatting.) |
Pi is 3.14 (talk | contribs) (→Video Solution) |
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+ | ==Problem== | ||
+ | |||
How many ordered pairs <math>(a, b)</math> of positive integers satisfy the equation | How many ordered pairs <math>(a, b)</math> of positive integers satisfy the equation | ||
<cmath>a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),</cmath> | <cmath>a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),</cmath> | ||
Line 7: | Line 9: | ||
==Solution== | ==Solution== | ||
− | Let <math>x = \lcm(a, b)</math>, and <math>y = \gcd(a, b)</math>. Therefore, <math>a\cdot b = | + | Let <math>x = \text{lcm}(a, b)</math>, and <math>y = \text{gcd}(a, b)</math>. Therefore, <math>a\cdot b = lcm(a, b)\cdot gcd(a, b) = x\cdot y</math>. Thus, the equation becomes |
<cmath>x\cdot y + 63 = 20x + 12y</cmath> | <cmath>x\cdot y + 63 = 20x + 12y</cmath> | ||
<cmath>x\cdot y - 20x - 12y + 63 = 0</cmath> | <cmath>x\cdot y - 20x - 12y + 63 = 0</cmath> | ||
− | Using Simon's Favorite Factoring Trick, we rewrite this equation as | + | Using [[Simon's Favorite Factoring Trick]], we rewrite this equation as |
<cmath>(x - 12)(y - 20) - 240 + 63 = 0</cmath> | <cmath>(x - 12)(y - 20) - 240 + 63 = 0</cmath> | ||
<cmath>(x - 12)(y - 20) = 177</cmath> | <cmath>(x - 12)(y - 20) = 177</cmath> | ||
− | Since <math>177 = 3\cdot 59</math> and <math>x > y</math>, we have <math>x - 12 = 59</math> and <math>y - 20 = 3</math>, or <math>x - 12 = 177</math> and <math>y - 20 = 1</math>. This gives us the solutions <math>(71, 23)</math> and <math>(189, 21)</math>. | + | Since <math>177 = 3\cdot 59</math> and <math>x > y</math>, we have <math>x - 12 = 59</math> and <math>y - 20 = 3</math>, or <math>x - 12 = 177</math> and <math>y - 20 = 1</math>. This gives us the solutions <math>(71, 23)</math> and <math>(189, 21)</math>. Since the <math>\text{GCD}</math> must be a divisor of the <math>\text{LCM}</math>, the first pair does not work. Assume <math>a>b</math>. We must have <math>a = 21 \cdot 9</math> and <math>b = 21</math>, and we could then have <math>a<b</math>, so there are <math>\boxed{\textbf{(B)} ~2}</math> solutions. |
(awesomeag) | (awesomeag) | ||
− | Edited by IronicNinja~ | + | Edited by IronicNinja, Firebolt360, and mprincess0229~ |
+ | |||
+ | == Video Solution by OmegaLearn== | ||
+ | https://youtu.be/zfChnbMGLVQ?t=494 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://www.youtube.com/watch?v=JWGHYUeOx-k | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2018|ab=B|num-b=22|num-a=24}} | {{AMC10 box|year=2018|ab=B|num-b=22|num-a=24}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 04:18, 21 January 2023
Problem
How many ordered pairs of positive integers satisfy the equation
where
denotes the greatest common divisor of
and
, and
denotes their least common multiple?
Solution
Let , and
. Therefore,
. Thus, the equation becomes
Using Simon's Favorite Factoring Trick, we rewrite this equation as
Since and
, we have
and
, or
and
. This gives us the solutions
and
. Since the
must be a divisor of the
, the first pair does not work. Assume
. We must have
and
, and we could then have
, so there are
solutions.
(awesomeag)
Edited by IronicNinja, Firebolt360, and mprincess0229~
Video Solution by OmegaLearn
https://youtu.be/zfChnbMGLVQ?t=494
~ pi_is_3.14
Video Solution
https://www.youtube.com/watch?v=JWGHYUeOx-k
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.