Difference between revisions of "2016 AMC 8 Problems/Problem 20"
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− | 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>, algebraically, it's some multiple of <math>b</math> and from looking at the numbers, we are sure that it is 3, thus b 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{A}</math> | + | 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>, algebraically, it's some multiple of <math>b</math> and from looking at the numbers, we are sure that it is 3, thus <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{A}</math> |
{{AMC8 box|year=2016|num-b=19|num-a=21}} | {{AMC8 box|year=2016|num-b=19|num-a=21}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 13:03, 23 November 2016
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 , algebraically, it's some multiple of and from looking at the numbers, we are sure that it is 3, thus 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
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 |
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