Clock Arithmetic & Modular Explorer

Analog Clock Dial (ℤ_6) Pointer: 1

Hand / Remainder (r)

Wrap Trail

Starting Base

Full Wraps (q)

Modulus m (2 to 36): 6

Arithmetic Parameters & Proof x = q × m + r

Quora Discussion Presets:

Base Integer (x)

Modulus (m)

Add/Subtract Offset (Δ)

Source Expression: x = 7, m = 6

Canonical Equation: 7 = 1 mod 6 (7 = 1 + (1)(6))

Clock Dial Position (r ∈ ℤ_m): 1

Full Laps / Quotient (q): 2

Remainder after offset (R): 1

Modular Inverse of (x mod m) in ℤ_m: N/A (gcd != 1)

Coprime Status: gcd(x, m) = 1

Evaluating: 7 mod 6 → Hand starts at 0, wraps clockwise 1 full lap (6 units), lands at position 1. Adding offset 6 → total 13 = 2 full laps + remainder 1.

Activity & Evaluation Log:

Init: x=7, m=6, offset=6Pos: 1

Whole laps and modular inverses

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Read the explanation

The saved clock uses residues zero through five when the modulus is six. Seven equals one plus one complete lap of six, so the pointer rests at one. Thirteen equals one plus two laps of six and lands at the same residue. The two bars show the actual lengths seven and thirteen, with lap boundaries every six units. Adding a full modulus changes the quotient while preserving the residue. Negative inputs need a nonnegative remainder. Minus one divided by six has floor quotient minus one, so minus one equals five minus six. Its residue is five. The source first takes the language remainder, adds the modulus, and reduces again. Base pointer and result remainder are different fields: starting at seven and adding six keeps both at one, while adding two gives result three without changing the base pointer field. A modular inverse reverses multiplication, not addition. Modulo six, five times five is twenty-five, which leaves remainder one. Therefore five is its own inverse. Six has greatest common divisor six with the modulus and has no inverse. The source runs the extended Euclidean algorithm and reports the inverse only when the greatest common divisor is one. These paired lengths twenty-five and twenty-four show the one-unit remainder.