Difference between revisions of "2011 AMC 12B Problems/Problem 5"
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<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math> | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math> | ||
==Solution== | ==Solution== | ||
− | <math>N</math> must be divisible by every positive integer less than <math>7</math>, or <math>1, 2, 3, 4, 5,</math> and <math>6</math>. Each number that is divisible by each of these | + | <math>N</math> must be divisible by every positive integer less than <math>7</math>, or <math>1, 2, 3, 4, 5,</math> and <math>6</math>. Each number that is divisible by each of these is a multiple of their least common multiple. <math>LCM(1,2,3,4,5,6)=60</math>, so each number divisible by these is a multiple of <math>60</math>. The smallest multiple of <math>60</math> is <math>60</math>, so the second smallest multiple of <math>60</math> is <math>2\times60=120</math>. Therefore, the sum of the digits of <math>N</math> is <math>1+2+0=\boxed{3\ \textbf{(A)}}</math> |
==See also== | ==See also== | ||
{{AMC12 box|year=2011|num-b=4|num-a=6|ab=B}} | {{AMC12 box|year=2011|num-b=4|num-a=6|ab=B}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 18:40, 7 August 2023
Problem
Let be the second smallest positive integer that is divisible by every positive integer less than . What is the sum of the digits of ?
Solution
must be divisible by every positive integer less than , or and . Each number that is divisible by each of these is a multiple of their least common multiple. , so each number divisible by these is a multiple of . The smallest multiple of is , so the second smallest multiple of is . Therefore, the sum of the digits of is
See also
2011 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 4 |
Followed by Problem 6 |
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 12 Problems and Solutions |
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