Chinese Remainder Theorem — Modular Rings (2D)

Coprime Moduli

Ring A

Ring B

Ring C

Counting speed 2.0 steps/s

Shared Counter

N (counter)0

N mod A0

N mod B0

N mod C0

Target residues

(0,0,0)

Modulus product M

Unique solution x (mod M)

How it works

Pick pairwise-coprime moduli m₁, m₂, m₃ (here always distinct primes, so coprimality is automatic). The Chinese Remainder Theorem says the system

x ≡ a₁ (mod m₁)
x ≡ a₂ (mod m₂)
x ≡ a₃ (mod m₃)

has exactly one solution x in the range [0, M), where M = m₁·m₂·m₃. It is built explicitly with the extended Euclidean algorithm:

Mᵢ = M / mᵢ
yᵢ = Mᵢ⁻¹  (mod mᵢ)      ← modular inverse via extended gcd
x  = Σ aᵢ · Mᵢ · yᵢ  (mod M)

Rings A/B/C — choose the sector count (modulus) of each concentric dial from a list of small primes. Each ring rotates independently in the plane, like a combination-lock wheel, so tooth N mod mᵢ sits under the fixed pointer at 12 o'clock — exactly the residue of the shared counter N.
Number-line trace below the dials plots N growing to the right; a colored tick lights up every time N passes a multiple of a modulus, and a gold marker shows where the CRT solution x — and every x + k·M after it — falls.
New Puzzle picks a random target residue for each ring and pre-computes the unique x via the formula above; Play/Step advance N and the rings turn until all three pointers land on their highlighted target sectors at once — that moment is N mod M = x, which the number-line trace marks explicitly.
Real-world relevance: this exact construction underlies fast RSA decryption (CRT-RSA), calendar/clock arithmetic, and error-correcting residue number systems used in some hardware arithmetic units.

Current sector (N mod m) Target sector Both at once (match)