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Excel in math and science
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It's hard to learn from lectures and videos
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Linear Diophantine Equations
What amounts of money can you make with only $5 and $7 bills? Can you make $23 with those 2 types of bills?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Concept Quizzes
A family restaurant sells nuggets in packages of two sizes, 7 nuggets and 12 nuggets. Is there a maximum number of nuggets that \( \color{red}{\text{cannot}} \) be expressed as a nonnegative combination of these package sizes?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Yes or No?
Are there any non-negative integer solutions to the equation
\[ 13m + 15n = 170? \]
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Yes or No?
If \( a>4 \) is an integer that is \( \color{red}{\text{not}} \) a multiple of 13, can we express \( 11a-39 \) as a non-negative integer linear combination of \( a \) and 13?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
In a fast food restaurant, you can order buckets of doughnuts only in quantities of 7 and 19 a bucket. What is the \( \color{red}{\text{second}} \) largest number of doughnuts you \( \color{red}{\text{cannot}} \) buy?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
There are only two kinds of postage stamps: 12 cents and 25 cents. For some values, you can get them as linear combinations of two kinds. For others, you cannot. What is the largest possible value you \( \color{red}{\text{cannot}} \) attain no matter which combination you choose? Give your answer in cents.