Difference between revisions of "2016 AMC 8 Problems/Problem 20"
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| + | ==Problem== | ||
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The least common multiple of <math>a</math> and <math>b</math> is <math>12</math>, and the least common multiple of <math>b</math> and <math>c</math> is <math>15</math>. What is the least possible value of the least common multiple of <math>a</math> and <math>c</math>? | The least common multiple of <math>a</math> and <math>b</math> is <math>12</math>, and the least common multiple of <math>b</math> and <math>c</math> is <math>15</math>. What is the least possible value of the least common multiple of <math>a</math> and <math>c</math>? | ||
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==Solution== | ==Solution== | ||
| − | {{ | + | We wish to find possible values of <math>a</math>, <math>b</math>, and <math>c</math>. By finding the greatest common factor of <math>12</math> and <math>15</math>, we can find that <math>b</math> is 3. Moving on to <math>a</math> and <math>c</math>, in order to minimize them, we wish to find the least such that the least common multiple of <math>a</math> and <math>3</math> is <math>12</math>, <math>\rightarrow 4</math>. Similarly, with <math>3</math> and <math>c</math>, we obtain <math>5</math>. The least common multiple of <math>4</math> and <math>5</math> is <math>20 \rightarrow \boxed{\textbf{(A)} 20}</math> |
| − | {{AMC8 box|year=2016|num-b= | + | |
| + | ==Video Solution by Pi Academy== | ||
| + | https://youtu.be/wvRmxjwOUHY?si=mNtAIGDHVdPaKWUX | ||
| + | |||
| + | |||
| + | ==Video Solution(CREATIVE THINKING + ANALYSIS!!!)== | ||
| + | https://youtu.be/_-xC-qQMCbk | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | |||
| + | == Video Solution by OmegaLearn== | ||
| + | https://youtu.be/HISL2-N5NVg?t=2340 | ||
| + | |||
| + | ~ pi_is_3.14 | ||
| + | |||
| + | == Video Solution == | ||
| + | https://youtu.be/7tGFq07njVo | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC8 box|year=2016|num-b=19|num-a=21}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 10:00, 9 November 2024
Contents
Problem
The least common multiple of 
and 
is 
, and the least common multiple of and 
is 
. What is the least possible value of the least common multiple of and ?
Solution
We wish to find possible values of , , and . By finding the greatest common factor of and , we can find that is 3. Moving on to and , in order to minimize them, we wish to find the least such that the least common multiple of and 
is , 
. Similarly, with and , we obtain 
. The least common multiple of 
and is 
Video Solution by Pi Academy
https://youtu.be/wvRmxjwOUHY?si=mNtAIGDHVdPaKWUX
Video Solution(CREATIVE THINKING + ANALYSIS!!!)
~Education, the Study of Everything
Video Solution by OmegaLearn
https://youtu.be/HISL2-N5NVg?t=2340
~ pi_is_3.14
Video Solution
~savannahsolver
See Also
| 2016 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
