Negative Modulo Calculator

Inputs

Result

r = 0 q = 0

Congruence

a ≡ r (mod m)

Results Table

#	a	m	Convention	q	r	Congruence

Formula Used

All conventions satisfy the integer division identity a = q·m + r. The conventions differ in how q is chosen, which controls the sign/range of r.

Euclidean: uses modulus |m|, chooses q = ⌊a/|m|⌋, giving 0 ≤ r < |m|. Implementation trick: r = ((a % |m|) + |m|) % |m|.
Truncated toward zero: chooses q = trunc(a/m) (e.g., intdiv), so r may be negative when a is negative.
Floor division remainder: chooses q = ⌊a/m⌋, so r can be non‑negative or have the sign of m depending on inputs.

For modular arithmetic in number theory, the Euclidean convention is standard: report r ∈ {0,1,…,|m|−1} and write a ≡ r (mod |m|).

How to Use This Calculator

Enter an integer dividend a and a non‑zero modulus m.
Select a convention: Euclidean (recommended), Truncated, or Floor.
Click Calculate to see q, r, steps, and congruence.
Use Add to table to accumulate multiple calculations.
Export the table as CSV or as a PDF document.

Tip: For negative moduli, Euclidean mode automatically uses |m| and reports a non‑negative residue.

Example Data Table

Click any row to load its a, m, and convention into the form.

a	m	Convention	q	r	Congruence
-13	5	Euclidean	-3	2	-13 ≡ 2 (mod 5)
-13	5	Truncated	-2	-3	-13 ≡ -3 (mod 5)
-13	5	Floor	-3	2	-13 ≡ 2 (mod 5)
-7	3	Euclidean	-3	2	-7 ≡ 2 (mod 3)
-7	3	Truncated	-2	-1	-7 ≡ -1 (mod 3)
-7	-3	Euclidean	-3	2	-7 ≡ 2 (mod 3)
13	-5	Truncated	-2	3	13 ≡ 3 (mod -5)
13	5	Euclidean	2	3	13 ≡ 3 (mod 5)

Why Different Modulo Conventions Exist

All conventions satisfy a = q·m + r, but each chooses q differently, changing the reported r. This matters when a or m is negative.

Number Theory: prefers non‑negative residues to represent congruence classes cleanly.
Systems Programming: truncated division matches legacy machine instructions and is fast.
Mathematical Analysis: floor division aligns with inequalities and interval reasoning.

When doing modular arithmetic proofs or cryptography, use the Euclidean convention so residues lie in [0, |m| − 1].

Mapping Between Conventions

Given any remainder r_x from a chosen convention and non‑zero m, the Euclidean residue is:

r_e = ((r_x % |m|) + |m|) % |m|

Direct formulas for each convention:

Truncated: q_t = trunc(a/m), r_t = a − q_t·m.
Floor: q_f = ⌊a/m⌋, r_f = a − q_f·m.
Euclidean: q_e = ⌊a/|m|⌋, r_e = a − q_e·|m|, with 0 ≤ r_e < |m|.

Example mapping with a = −13, m = 5:

Convention	q	r	Note
Truncated	q_t = −2	r_t = −3	May return negative remainder.
Floor	q_f = −3	r_f = 2	Remainder non‑negative for m > 0.
Euclidean	q_e = −3	r_e = 2	Residue in [0,4].

From r_t = −3, compute r_e = ((−3 % 5)+5)%5 = 2.

Common Pitfalls and Quick Checks

Never use m = 0; the modulus must be non‑zero.
For Euclidean residues, always verify 0 ≤ r < |m|.
Confirm the identity a = q·m + r numerically after each computation.
Mixing conventions can confuse tests; normalize to Euclidean for comparisons.
Negative m? Use |m| when reporting residues in Euclidean mode.

If portability matters, store normalized Euclidean residues and the original m alongside results.