Mathematical Induction Divisibility Calculator

Calculator Input

Claim name

Expression f(n)

Divisor d

Start n

End n

Step samples

Allowed input includes n, numbers, +, -, *, /, %, ^, parentheses, abs(), and sqrt(). Use * for multiplication when unsure.

Example Data Table

Expression	Divisor	Start n	End n	Reason to test
n^3-n	6	1	12	Classic product of three consecutive integers.
n^5-n	5	0	15	Useful modular pattern for powers.
2^(2*n)-1	3	1	10	Shows geometric power divisibility.
n^2+n	2	1	14	Tests product of consecutive integers.

Formula Used

A divisibility claim uses d | f(n). This means f(n) is a multiple of d.

Remainder test: f(n) mod d = 0.

Base case: f(n0) mod d = 0.

Induction assumption: f(k) = d × q for some integer q.

Difference method: if f(k+1) - f(k) = d × m, then f(k+1) = d × (q + m).

Normalized remainder: ((value mod |d|) + |d|) mod |d|.

How to Use This Calculator

Enter the expression in n. Then enter the divisor.

Choose the first n for the base case. Then choose the last n for testing.

Enter how many sample induction steps you want to inspect.

Press Calculate. The result appears above the form.

Review the remainder table. A zero remainder means that row passes.

Download the CSV or PDF when you need a record.

Understanding Induction Divisibility

Mathematical induction proves statements across all whole numbers in a range. A divisibility claim often says one expression is always a multiple of another number. This calculator helps test that claim before you write the proof. It checks the base case, evaluates many values, and studies the induction step. It also gives a proof outline that you can refine.

Why the Base Case Matters

The base case is the first allowed value. If the base case fails, the full claim fails. For example, a claim may start at n equals one. The calculator evaluates that exact starting value. It reports the value, remainder, and pass status. This makes errors easy to spot early.

How the Step Check Helps

Induction needs a bridge from k to k plus one. A common divisibility method uses the difference. If f(k) is divisible by d, make an assumption. If the difference is divisible by d, the next case follows. The calculator samples that difference across your selected range. It shows the step value and its remainder. This helps you see a useful proof pattern.

Reading the Remainder Table

The table lists each tested n. It shows f(n), the normalized remainder, and a decision. A zero remainder means divisibility holds for that row. A nonzero remainder marks a counterexample. The period note can reveal repeating remainder cycles. That is useful for modular arithmetic proofs.

Best Use Cases

Use this tool for expressions like n cubed minus n. It also works for powers, products, sums, and quotients that produce integers. Keep the range moderate when powers grow quickly. Very large values can exceed normal server limits. Try several starting values when the exact domain is uncertain. Record each counterexample before editing the claim. Use the exported table for homework notes. Use the generated proof note as a draft. Then add your own algebraic explanation. State the final domain in every proof clearly.

Proof Writing Reminder

This calculator supports proof building. It does not replace exact reasoning. A correct induction proof still needs a clear assumption. It also needs a symbolic step. Use the numeric checks to find structure. Then write why the structure always works. That balance makes your proof stronger.

FAQs

What does this calculator prove?

It tests values and creates an induction proof outline. A final proof still needs algebra that shows the step works for every valid integer.

What expression format should I use?

Use n as the variable. Operators include +, -, *, /, %, and ^. Parentheses, abs(), and sqrt() are also supported.

Why does my result show not integer?

The evaluated expression produced a decimal result. Divisibility usually needs integer values, so revise the expression or starting range.

Can I use negative divisors?

Yes. The calculator uses the absolute value of the divisor when normalizing remainders, so the divisibility decision remains consistent.

What is the base case?

The base case is the first n in the claim. If it fails, induction cannot prove the claim from that starting value.

What does the step table show?

It compares f(k), f(k+1), and their difference. A zero difference remainder can reveal a useful induction strategy.

Does a passing table guarantee proof?

No. Many passing rows give strong evidence, but a proof needs a symbolic argument for all allowed values.

Why limit the range?

Large ranges and high powers can create heavy calculations. The limit keeps the page responsive and easier to export.