Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-16
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λ(561)=80 and every integer coprime to 561 has eightieth power congruent to one

Example

One has λ(561)=80 and φ(561)=320. Consequently every a coprime to 561 satisfies a80≡1(mod561).

Facts & Assumptions

Given: The factorisation 561=3⋅11⋅17.

[L1]

Carmichael's function is the least common multiple of its prime-power values (Carmichael's function on prime powers and its least-common-multiple formula).

[L2]

If gcd⁡(a,n)=1, then aλ(n)≡1(modn) (If gcd⁡(a,n)=1, then aλ(n)≡1(modn)).

[L3]

Euler's totient is multiplicative on coprime arguments (Euler's totient is multiplicative: gcd⁡(m,n)=1 implies φ(mn)=φ(m)φ(n) for positive m,n).

Verification

technique · direct
1.1L1L3algebra

By [L1], λ(561)=lcm⁡(2,10,16)=80, while [L3] gives φ(561)=2⋅10⋅16=320.

2.1step 1.1L2∎

Apply [L2] to the value in step 1.1 to obtain a80≡1(mod561) for every a coprime to 561.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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.