Linear Diophantine Equations
A family restaurant sells nuggets in packages of two sizes, 7 nuggets and 12 nuggets. Is there a maximum number of nuggets that be expressed as a nonnegative combination of these package sizes?
Yes or No?
Are there any non-negative integer solutions to the equation
Yes or No?
If is an integer that is a multiple of 13, can we express as a non-negative integer linear combination of 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 largest number of doughnuts you 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 attain no matter which combination you choose? Give your answer in cents.