Modulo Of Negative Numbers - Floor, Truncated, and Euclidean

The modulo of negative numbers calculator is a focused math tool that takes any integer dividend and divisor and shows the remainder using three common conventions: the floored remainder used in mathematics and Python, the truncated remainder used in C, Java, and JavaScript, and the Euclidean remainder that is always non-negative. Most calculators only show one of these answers, which is why -7 mod 3 looks wrong to people who switch between languages. This page shows all three at once so the sign rule is visible at a glance.

• • Translating between languages: Move an expression from Python to JavaScript (or back) without surprising negative remainders, by reading both conventions side by side.
• • Verifying clock arithmetic: Check 12-hour clock math where the dividend can wrap negative; confirm the floored remainder matches the clock hand.
• • Debugging array indexing: When an index is allowed to go negative, the floored vs truncated distinction decides whether -1 wraps to the last element or returns out of range.
• • Homework and exam practice: Compute -12 mod 5, -3 mod 7, or 7 mod -3 by hand and confirm against the quotient and remainder.

Modular arithmetic is built around remainders, and the only real disagreement between definitions is what to do when the dividend or divisor is negative. Math and Python choose the floored definition so the result lands in [0, n) for a positive divisor. C, Java, and JavaScript choose the truncated definition, putting -7 mod 3 at -1 instead of 2. Both are correct.

If both numbers are positive, the Modulo Calculator handles the basic remainder, modular inverse, and modular exponentiation in one form.

The calculator implements all three definitions directly so the numbers you see match what you would compute by hand. What changes between the three modes is which integer quotient k is paired with the dividend; the remainder follows from there.

• a: The dividend. Any integer, including negative values.
• n: The divisor. Any non-zero integer, including negative values.
• k: The integer quotient chosen by the convention. Floor for Python, trunc for C/JS, and a value that keeps r non-negative for Euclidean.
• r: The remainder. Always satisfies a = n*k + r, and the range depends on the chosen convention.
• mode: The dropdown. 'floored' uses floor, 'truncated' uses trunc, and 'euclidean' uses the always-non-negative variant.

The output panel reports the primary remainder, the truncated remainder, the integer quotient, and a one-line sign rule. Floored or Euclidean gives a non-negative answer for a positive divisor; Truncated keeps the sign of the dividend, matching what C or JavaScript would print.

For negative divisors, the floored convention puts the remainder in (n, 0]. The calculator handles that automatically, so 7 mod -3 shows primary remainder -2 and integer quotient -3.

According to Wikipedia, the floored variant of the modulo operation returns a remainder in [0, n) for a positive divisor, while the truncated variant used in C, Java, and JavaScript keeps the sign of the dividend.

Once you have a remainder for a single mod step, the Is Modulo Associative Calculator shows whether chaining (a mod n) mod m gives the same answer as a mod (n mod m) - a question that only shows up once you start composing these operations.

Four small ideas explain every behavior you see on this modulo of negative numbers calculator. Keeping them straight is the difference between guessing the sign and predicting it.

Most negative-remainder surprises come from swapping languages without changing definitions. The integer quotient k is where to look first: floored and truncated pick a different k for the same a and n, and that single choice flips the sign when the dividend is negative.

The Quotient Calculator computes integer quotients for any pair of integers, which is useful when you want to verify k by hand before trusting a mod result.

Three inputs produce all three remainders. Type the dividend, choose the divisor, and pick the convention you want as the primary answer.

1. 1 Enter the dividend: Type the dividend (a). Negative values are allowed and encouraged, since this calculator exists specifically to handle them.
2. 2 Enter the divisor: Type the divisor (n). Zero is rejected with a clear error so you cannot accidentally compute a mod 0.
3. 3 Pick the convention: Use the dropdown to choose Floored, Truncated, or Euclidean. The default is Floored so the page matches Python and standard math notation.
4. 4 Read the primary remainder: Look at the primary value first. This is the answer under the convention you picked.
5. 5 Compare with the truncated value: Glance at the truncated remainder row to see what C, Java, or JavaScript would print for the same inputs. Use it as a sanity check when porting code.
6. 6 Verify by hand with the quotient: Multiply the divisor by the integer quotient, add the primary remainder, and confirm you get the dividend back. That single check covers sign, magnitude, and convention at once.

Try -7 mod 3 with the default Floored convention: dividend -7, divisor 3, dropdown on Floored. The primary remainder reads 2, the truncated remainder -1, and the integer quotient -3. Switch to Truncated and the primary flips to -1 with quotient -2. Now switch a to 7 and n to -3: the floored remainder drops to -2 with quotient -3, and the truncated column reads 1 because 7 - (-3)(-2) = 1.

When you only need the truncated sign-of-dividend answer and not the floored or Euclidean views, the Remainder Calculator gives you that one number with a clean division-style layout that mirrors long-division homework.

Negative modulo wastes an afternoon when the sign is wrong. Putting all three conventions on one page turns it into a five-second check.

• • Three conventions at a glance: Floored, truncated, and Euclidean answers sit next to each other so the sign rule is visible, not memorized.
• • Quotient for hand verification: Divisor times quotient plus remainder should equal the dividend; the panel shows k so the check is one multiplication away.
• • Negative divisors handled correctly: 1 mod -5 and 7 mod -3 produce the right floored and truncated split without any extra work.
• • Safer language ports: A side-by-side comparison of the floored and truncated answer removes the most common Python-to-JavaScript (or back) porting bug.

The biggest win is dropping the language-by-language memorization step. Once you have seen the floored and truncated answers for -7 mod 3, the same approach works for any negative-dividend case.

If the negative-modulo question turns out to be one step inside a system of congruences x congruent to a mod n, the Chinese Remainder Calculator solves the whole system and reports the canonical x, so you do not have to compute the remainders one at a time.

Five things decide what the calculator prints. Knowing them keeps the answers predictable.

• • The calculator works only on integers. Floating-point dividends like -7.5 mod 3 are not supported; use a dedicated floating-point tool instead.
• • Truncated mode follows the sign-of-dividend rule used by C, Java, and JavaScript. It does not match languages that define modulo differently, so double-check the sign rule for the target system.

Even with all three answers visible, the only definitive answer is the one tied to a specific language. If you need the JavaScript answer, the truncated column gives you -1 without leaving the browser.

The sign of the divisor and dividend together decide the interval. For positive n, floored and Euclidean answers always sit in [0, n); for negative n, the floored interval becomes (n, 0].

According to Python documentation, the % operator uses floored division, so -7 % 3 evaluates to 2 and 7 % -3 evaluates to -2.

According to MDN, JavaScript's remainder operator % is defined so the result carries the sign of the dividend, which is why -7 % 3 evaluates to -1 in any browser.

Beyond sign conventions, the Uses Of Modulo Calculator lists where negative modulo actually shows up in practice - hash tables, clock arithmetic, calendar offsets, and the wrap-around logic that depends on the floored vs truncated rule this page exposes.