ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-07-31

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The system $x \equiv 2 (m o d 3)$ , $x \equiv 3 (m o d 5)$ , $x \equiv 2 (m o d 7)$ has the unique solution $x \equiv 23 (m o d 105)$

Example

The system

$x \equiv 2 (m o d 3), x \equiv 3 (m o d 5), x \equiv 2 (m o d 7)$

has exactly the solutions $x \equiv 23 (m o d 105)$ .

Facts & Assumptions

Given: The three displayed congruences.

[L1]

For a finite pairwise-coprime list of positive moduli, prescribed residues determine one class modulo their product (Chinese remainder theorem for a finite pairwise-coprime list: simultaneous residues determine one class modulo the product, and the resulting bijection preserves addition and multiplication).

[L2]

The relation $u \equiv v (m o d n)$ means $n ∣ (u - v)$ (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

Verification

technique · constructive

1.1

The numbers $3, 5, 7$ are pairwise coprime and have product $105$ . The complementary products $35, 21, 15$ satisfy $2 \cdot 35 \equiv 1 (m o d 3)$ , $21 \equiv 1 (m o d 5)$ , and $15 \equiv 1 (m o d 7)$ , while each is divisible by the other two moduli.

2.1

Therefore $x : = 2 (2 \cdot 35) + 3 (21) + 2 (15) = 233$ has the prescribed three residues, and $233 = 23 + 2 \cdot 105$ gives $x \equiv 23 (m o d 105)$ .

constructstep 1.1L2

3.1

By [L1], all simultaneous solutions form one class modulo $3 \cdot 5 \cdot 7 = 105$ ; since step 2.1 exhibits $23$ in that class, it is exactly $[23]_{105}$ .

step 2.1L1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Conrad, The Chinese Remainder Theorem (standard reference, not scraped)
University of Southampton, Simultaneous Linear Congruences (standard reference, not scraped)

The system x≡2(mod3)x≡2(mod3), x≡3(mod5)x≡3(mod5), x≡2(mod7)x≡2(mod7) has the unique solution x≡23(mod105)x≡23(mod105)