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# Puzzles

## Combining multiples

In each of these questions, positive integers should be taken to include 0.
1. What is the largest number that cannot be written in the form $$3a+5b$$, where $$a$$ and $$b$$ are positive integers?
2. What is the largest number that cannot be written in the form $$3a+7b$$, where $$a$$ and $$b$$ are positive integers?
3. What is the largest number that cannot be written in the form $$10a+11b$$, where $$a$$ and $$b$$ are positive integers?
4. Given $$n$$ and $$m$$, what is the largest number that cannot be written in the form $$na+mb$$, where $$a$$ and $$b$$ are positive integers?
If you enjoyed this puzzle, check out Sunday Afternoon Maths LIV,
puzzles about numbers, or a random puzzle.

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