Overview

Some number theory problems do not fit one single formula category. This module collects practical integer techniques that appear frequently in contest problems.

Key Ideas

• Palindrome and digit-structure questions are often handled by place value and modular checks.
• Coin and payment questions are usually linear Diophantine equations with nonnegative constraints.
• Integer-operation conditions can often be solved by parity and remainder analysis.
• For largest nonrepresentable coin values with coprime denominations $a,b$, use the Chicken McNugget result: $ab-a-b$.

Core Skills

Digit Structure with Constraints

If a number is written as $100a+10b+c$, convert verbal conditions into equations in $a,b,c$ with digit bounds.

Two-Coin Representation

For values representable as $ax+by=n$ with $x,y\ge0$, first solve in integers, then apply nonnegative bounds.

Parity and Mod Checks

If an equation has no solution by parity or mod class, eliminate it early before expanding cases.

Worked Example

What is the largest integer that cannot be written as $4x+7y$ with nonnegative integers $x,y$?

Since $\gcd(4,7)=1$, the largest nonrepresentable value is

So the answer is $\boxed{17}$.

Practice Problems