Difference between revisions of "2018 AMC 8 Problems/Problem 21"
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==Problem 21== | ==Problem 21== | ||
− | How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by | + | How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by II? |
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<math>\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5</math> | <math>\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5</math> | ||
Revision as of 19:00, 1 November 2020
Problem 21
How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by II?
Video Solution
https://youtu.be/7an5wU9Q5hk?t=939
Solution 1
Looking at the values, we notice that ,
and
. This means we are looking for a value that is four less than a multiple of
,
, and
. The least common multiple of these numbers is
, so the numbers that fulfill this can be written as
, where
is a positive integer. This value is only a three digit integer when
is
or
, which gives
and
respectively. Thus we have
values, so our answer is
Video Solution
See Also
2018 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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.