What is -7 mod 3, exactly?

It depends on the convention, and that is the whole point. The % operator in JavaScript, C, Java, Go and Rust returns -1, because those languages make the remainder take the sign of the dividend (-7). The mathematical, or Euclidean, mod returns 2, because by definition the result must land in [0, n) — and -7 = (-3)·3 + 2. This calculator shows both so you can match whichever your code or your textbook expects. Python's % actually returns 2 here, because Python uses the Euclidean-style sign for a positive modulus, which is why ported C code so often breaks on negative inputs.

What is the difference between the programming % and the math mod?

They agree for every non-negative dividend and only diverge when the dividend is negative. The % operator (truncated remainder) is defined as a - trunc(a/n)·n, so its result carries the sign of a. The math mod (Euclidean) is defined as a - floor(a/n)·n, so its result is never negative. Concretely, -1 % 256 is -1 in C but the byte value you usually want is 255 — that is the Euclidean answer. If you are indexing into an array or wrapping a clock, you almost always want the Euclidean result.

When does a modular inverse exist?

The modular inverse of a mod n exists if and only if a and n are coprime, i.e. gcd(a, n) = 1. When it exists, it is the unique x in [0, n) with a·x ≡ 1 (mod n), and this tool finds it with the extended Euclidean algorithm. When gcd(a, n) ≠ 1 there is no such x, and the tool says so instead of returning a wrong number. Example: 3 has an inverse mod 11 (it is 4, since 3·4 = 12 ≡ 1), but 2 has no inverse mod 4 because they share the factor 2.

Why doesn't modular exponentiation overflow on a huge exponent?

Because it never computes aᵇ in full. Fast exponentiation (square-and-multiply) reduces modulo n after every squaring, so the running value stays smaller than n² the entire time — it grows logarithmically in the exponent, not exponentially. Computing 2^1000000 mod 1000000007 takes about 20 multiplications, each on numbers well under 60 bits. With BigInt underneath there is no fixed ceiling, so even a 40-digit modulus is exact, which is exactly the regime RSA and Diffie–Hellman operate in.

How large can the numbers be?

Up to 40 digits per operand, all handled with JavaScript BigInt so the arithmetic is exact — no floating-point rounding. The 40-digit cap is a sanity limit on input length, not a limit of the math; BigInt itself has no fixed size. The modulus n must be non-zero (division by zero is undefined), and the exponent for modular power must be non-negative.

Wrap an array index that can go negative

You move a cursor left through a list of length 8 and want it to wrap to the end: index -1 should become 7. In JavaScript -1 % 8 is -1, which throws or returns undefined when you index with it. Paste a = -1, n = 8 and read the Euclidean result: 7. That is the formula ((i % n) + n) % n your code needs, confirmed in one place instead of debugging an off-by-one wrap in production.

Compute a day of the week

Zeller-style date math lands you on "day number" 17 and you want it mod 7 to pick a weekday. For positive inputs both conventions agree (17 mod 7 = 3), but when an offset calculation goes negative — say -4 mod 7 — you need 3, not -4. The calculator shows the Euclidean 3 so your weekday lookup table never indexes out of bounds.

Hash into a bucket without negative slots

A string hash returns a signed 32-bit value that can be negative, and you bucket it into a table of 1009 slots with hash mod 1009. With the truncated % a negative hash gives a negative bucket index and a crash. Drop the hash and 1009 in here, take the Euclidean column, and you get a valid 0..1008 slot every time.

Find an RSA private exponent by hand

For a teaching RSA example with e = 17 and φ(n) = 3120, the private key d is the modular inverse of e mod φ(n). Enter a = 17, n = 3120 and the tool returns d = 2753 via the extended Euclidean algorithm, then you can sanity-check it with the modular power field: 17·2753 mod 3120 should be 1. No paper-and-pencil Euclidean back-substitution required.

Verify a modular exponentiation in a crypto exercise

A Diffie–Hellman walkthrough asks for 5^117 mod 23. Typing 5^117 into a normal calculator overflows or rounds; here you enter a = 5, n = 23, exponent 117 and read the exact answer instantly, because the tool uses square-and-multiply with BigInt and never forms the 80-digit power.

Modulo Calculator — truncated vs Euclidean mod, floor quotient, modular power & inverse

a mod n in both conventions — the % your language returns AND the math mod that is never negative, plus floor quotient, modular power and inverse

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Dividend a

Modulus n

Exponent b (optional, for aᵇ mod n)

Truncated remainder (a % n)

-1

The % operator in JS, C, Java, Go, Rust. Sign follows the dividend a.

Euclidean / math mod

Always in [0, |n|). What number theory, clocks and crypto use.

These two disagree here because a is negative: % keeps a’s sign, the math mod does not.

Floor quotient ⌊a / n⌋

-3

Check: -7 = (-3)·3 + 2

Pairs with the Euclidean mod: a = q·n + r.

Modular inverse a⁻¹ mod n

a⁻¹ ≡ 2 (mod n), because a·2 ≡ 1

What this tool does

A modulo calculator that finally answers the question that trips up every programmer the first time they hit a negative number: what is -7 mod 3? Your language's % operator says -1; a number theorist says 2. Both are correct under their own convention, and this tool computes BOTH side by side so you never guess again. The "truncated" result is what JavaScript, C, Java, Go and Rust return from %, where the sign follows the dividend. The "Euclidean" result is the mathematician's residue, always in the range [0, |n|), which is what clock arithmetic, ring-buffer indexing and cryptography actually want. Alongside the two remainders you get the floor quotient that makes the division identity a = q·n + r hold exactly, modular exponentiation aᵇ mod n by fast square-and-multiply (so 2^1000000 mod 97 is instant and never builds the giant intermediate), and the modular multiplicative inverse via the extended Euclidean algorithm — with a clear "no inverse exists" when a and n share a factor. Every value is computed with JavaScript BigInt, so 40-digit operands stay exact instead of silently rounding the way doubles would. Inputs round-trip through the URL, so a share link reproduces the exact calculation.

Tool details

Input: Form fields; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 10 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Developer; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Modulo Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Wrap an array index that can go negative
You move a cursor left through a list of length 8 and want it to wrap to the end: index -1 should become 7. In JavaScript -1 % 8 is -1, which throws or returns undefined when you index with it. Paste a = -1, n = 8 and read the Euclidean result: 7. That is the formula ((i % n) + n) % n your code needs, confirmed in one place instead of debugging an off-by-one wrap in production.
Compute a day of the week
Zeller-style date math lands you on "day number" 17 and you want it mod 7 to pick a weekday. For positive inputs both conventions agree (17 mod 7 = 3), but when an offset calculation goes negative — say -4 mod 7 — you need 3, not -4. The calculator shows the Euclidean 3 so your weekday lookup table never indexes out of bounds.
Hash into a bucket without negative slots
A string hash returns a signed 32-bit value that can be negative, and you bucket it into a table of 1009 slots with hash mod 1009. With the truncated % a negative hash gives a negative bucket index and a crash. Drop the hash and 1009 in here, take the Euclidean column, and you get a valid 0..1008 slot every time.
Find an RSA private exponent by hand
For a teaching RSA example with e = 17 and φ(n) = 3120, the private key d is the modular inverse of e mod φ(n). Enter a = 17, n = 3120 and the tool returns d = 2753 via the extended Euclidean algorithm, then you can sanity-check it with the modular power field: 17·2753 mod 3120 should be 1. No paper-and-pencil Euclidean back-substitution required.
Verify a modular exponentiation in a crypto exercise
A Diffie–Hellman walkthrough asks for 5^117 mod 23. Typing 5^117 into a normal calculator overflows or rounds; here you enter a = 5, n = 23, exponent 117 and read the exact answer instantly, because the tool uses square-and-multiply with BigInt and never forms the 80-digit power.

Common pitfalls

Assuming -7 % 3 is 2 in C-family languages. It is -1 — the % operator there takes the sign of the dividend. If you need the always-non-negative result, use the Euclidean column or write ((a % n) + n) % n in your code.
Indexing an array directly with a raw % result. A negative dividend gives a negative index in most languages, which is out of bounds. Wrap with the Euclidean mod before indexing a ring buffer or hash bucket.
Expecting every number to have a modular inverse. The inverse of a mod n exists only when gcd(a, n) = 1. If a and n share a factor there is no inverse, and any value you get from a careless formula is simply wrong.

Privacy

Every calculation — both remainders, the floor quotient, modular exponentiation and the extended Euclidean inverse — is plain JavaScript that runs in your browser tab. No number you enter ever leaves the page, nothing is logged, and there is no external API call. One thing to know: the shareable URL encodes your inputs in the query string (e.g. ?a=-7&n=3), so pasting a share link records those numbers in the destination server's access log. For homework that is harmless; if a value is sensitive — say an RSA private exponent — copy the result manually instead of sharing the URL.

FAQ

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Modulo Calculator — truncated vs Euclidean mod, floor quotient, modular power & inverse

Truncated remainder (a % n)

Euclidean / math mod

Floor quotient ⌊a / n⌋

Modular inverse a⁻¹ mod n

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Modulo Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Wrap an array index that can go negative

Compute a day of the week

Hash into a bucket without negative slots

Find an RSA private exponent by hand

Verify a modular exponentiation in a crypto exercise

Common pitfalls

Privacy

FAQ

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