Let a, b, c Z. Prove that if abc 1 (mod 3), then an odd number of a, b and c are

Step 1 of 3

Given \(a b c \equiv 1(\bmod 3) for some integers a, b and c.

It is known that for any positive integer m,

Consider the following two cases:

Case 1.1: If none of a, b and c is congruent to 1 modulo 3.

\(a b c \equiv 1(\bmod 3)\) 

Assume that at least one of a, b and c is congruent to 0 modulo 3, say \(a \equiv 0(\bmod 3)\).

Then \(a=3 q\) for some integer q.

Thus \(a b c=3 q b c\) 

Since \(q b c \in Z\), it follows that \(3 \mid a b c\) .

Then \(a b c=0(\bmod 3)\) which gives \(a b c \neq 1(\bmod 3)\), which contradicts the assumption.