Difference between revisions of "2012 AMC 10B Problems/Problem 4"
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− | + | == Problem 4 == | |
− | + | When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be leftover? | |
+ | |||
+ | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math> | ||
+ | |||
+ | |||
+ | == Solution 1 == | ||
+ | |||
+ | |||
+ | In total, there were <math>3+4=7</math> marbles left from both Ringo and Paul.We know that <math>7 \equiv 1 \pmod{6}</math>. This means that there would be <math>1</math> marble leftover, or <math>\boxed{A}</math>. | ||
+ | |||
+ | == Solution 2 (modulo) == | ||
+ | Let <math>r</math> be the number of marbles Ringo has and let <math>p</math> be the number of marbles Paul has. we have the following equations: | ||
+ | <cmath> r \equiv 4 \mod{6} </cmath> | ||
+ | <cmath> p \equiv 3 \mod{6} </cmath> | ||
+ | Adding these equations we get: | ||
+ | <cmath> p + r \equiv 7 \mod{6} </cmath> | ||
+ | We know that <math>7 \equiv 1 \mod{6}</math> so therefore: | ||
+ | <cmath> p + r \equiv 7 \equiv 1 \mod{6} \implies p + r \equiv 1 \mod{6} </cmath> | ||
+ | Thus when Ringo and Paul pool their marbles, they will have <math>\boxed{\textbf{(A)}\ 1}</math> marble left over. | ||
+ | |||
+ | ~ herobrine-india | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{AMC10 box|year=2012|ab=B|num-b=3|num-a=5}} | ||
+ | {{MAA Notice}} |
Latest revision as of 18:07, 13 July 2021
Problem 4
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be leftover?
Solution 1
In total, there were marbles left from both Ringo and Paul.We know that . This means that there would be marble leftover, or .
Solution 2 (modulo)
Let be the number of marbles Ringo has and let be the number of marbles Paul has. we have the following equations: Adding these equations we get: We know that so therefore: Thus when Ringo and Paul pool their marbles, they will have marble left over.
~ herobrine-india
See Also
2012 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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.