Congruence Modulo Integer/Examples/531 not equiv 1236 mod 7561

Example of Non-Congruence Modulo an Integer

$531 \not \equiv 1236 \pmod {7561}$

Proof

Aiming for a contradiction, suppose $531 \equiv 1236 \pmod {7561}$.

We have that:

\(\ds \size {531}\)

\(<\)

\(\ds \dfrac {7561} 2\)

\(\ds \size {1236}\)

\(<\)

\(\ds \dfrac {7561} 2\)

From Congruent Integers less than Half Modulus are Equal, it follows that $531 = 1236$.

The result follows by Proof by Contradiction.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Exercise $7 \ \text{(d)}$