Integer Equations and Place Value

An equation with integer unknowns has more structure than the same equation over the reals. Divisibility and digit restrictions can reduce an apparently endless search to a short complete list.

What you will learn

• Factor integer equations
• Bound solutions
• Model digit conditions

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Integer solutions require arithmetic structure

An integer equation asks for whole-number values, often with positivity or digit conditions as well. A linear equation may have infinitely many real solutions but only a few positive integer solutions. Reduce modulo one coefficient to constrain the other variable, then use positivity to limit its size. This combination gives a systematic search rather than guesswork.

Always distinguish positive integers from nonnegative integers. Zero can create extra boundary solutions, and negative values can greatly expand the set. Write the permitted range beside each variable before counting. After finding all candidates, substitute them into the original equation and explain why your bounds excluded every other possibility.

Factorization turns equations into divisor problems

When variables multiply, rearrange the equation to expose a product of integer factors. A fixed positive product has finitely many positive factor pairs, so divisor structure becomes a solution method. Sometimes adding a small constant to both sides completes a product. Expand your proposed factors to verify that the rearrangement preserves the equation.

Difference-of-squares equations are another example: $(x-y)(x+y)$ equals a fixed integer. The two factors must have the same parity for their half-sum and half-difference to be integers. Counting every factor pair without this parity check overcounts. Positivity conditions on $x,y$ can also eliminate equal factors or reverse their allowed order.

Digits are coefficients in a positional expansion

In base $b$, a numeral with digits $a,c,d$ represents $ab^2+cb+d$. Each digit lies between zero and $b-1$, and a leading digit is nonzero. A digit symbol is a value, while its position determines its weight. Reversing a numeral exchanges the weights, not just the names of the digits.

When a problem gives a relation between a numeral and its reversal, write both positional expressions before simplifying. The common terms often cancel, leaving a divisibility condition on digit differences. If the base is unknown, the largest digit supplies a lower bound on it. These restrictions are part of the mathematics, not merely notation.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Forgetting whether zero is allowed.
• Counting a factor pair whose parity cannot produce integer variables.
• Ignoring the largest digit when checking a possible base.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• Combine congruences with bounds to prove completeness.
• Use factorization to replace an equation by a finite divisor problem.
• Expand positional numerals and retain every digit restriction.