Number Theory

Linear Diophantine Equations

Postage Stamp Problem / Chicken McNugget Theorem

         

A family restaurant sells nuggets in packages of two sizes, 7 nuggets and 12 nuggets. Is there a maximum number of nuggets that cannot \color{#D61F06}{\text{cannot}} be expressed as a nonnegative combination of these package sizes?

Yes or No?

Are there any non-negative integer solutions to the equation

13m+15n=170? 13m + 15n = 170?

Yes or No?

If a>4 a>4 is an integer that is not \color{#D61F06}{\text{not}} a multiple of 13, can we express 11a39 11a-39 as a non-negative integer linear combination of a a and 13?

In a fast food restaurant, you can order buckets of doughnuts only in quantities of 7 and 19 a bucket. What is the second \color{#D61F06}{\text{second}} largest number of doughnuts you cannot \color{#D61F06}{\text{cannot}} buy?

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 cannot \color{#D61F06}{\text{cannot}} attain no matter which combination you choose? Give your answer in cents.

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