To prove that if \( b \) is coprime to 6, then \( b^2 \equiv 1 \mod 24 \), it is essential to establish that \( \text{gcd}(b, 6) = 1 \). This condition implies that \( b \) cannot have 2 or 3 as factors. Consequently, \( b \) can take on two possible values modulo 6, which can be expressed as \( b = 6n + r \) for integers \( n \) and \( r \). Expanding \( (6n + r)^2 \) leads to the conclusion that \( b^2 \equiv 1 \mod 24 \).
PREREQUISITESThis discussion is beneficial for students studying number theory, particularly those tackling modular arithmetic and proofs involving congruences. It is also useful for educators seeking to explain concepts of coprimality and gcd in a mathematical context.