Congruence of Quotient/Warning

Congruence of Quotient: Warning

Note that in general it is not the case that:

\(\ds a\)

\(\equiv\)

\(\ds b\)

\(\ds \pmod m\)

\(\ds c\)

\(\equiv\)

\(\ds d\)

\(\ds \pmod m\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac a c\)

\(\equiv\)

\(\ds \dfrac b d\)

For example:

\(\ds 8\)

\(\equiv\)

\(\ds 18\)

\(\ds \pmod m\)

\(\ds 27\)

\(\equiv\)

\(\ds 7\)

\(\ds \pmod m\)

But we may not conclude that:

$\dfrac 8 {27} \pmod {10} = \dfrac {18} 7 \pmod {10}$

We may not even conclude that:

$\dfrac 8 2 \pmod {10} = \dfrac {18} 2 \pmod {10}$

because:

\(\ds \dfrac 8 2 \pmod {10}\)

\(\equiv\)

\(\ds 4 \pmod {10}\)

\(\ds \dfrac {18} 2 \pmod {10}\)

\(\equiv\)

\(\ds 9 \pmod {10}\)

\(\ds \)

\(\not \equiv\)

\(\ds 4 \pmod {10}\)

But we do have:

\(\ds \dfrac 8 2 \pmod {10 / 2}\)

\(\equiv\)

\(\ds 4 \pmod 5\)

\(\ds \dfrac {18} 2 \pmod {10 / 2}\)

\(\equiv\)

\(\ds 9 \pmod 5\)

\(\ds \)

\(\equiv\)

\(\ds 4 \pmod 5\)

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): congruence modulo $n$ (C.F. Gauss, 1801)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): congruence modulo $n$ (C.F. Gauss, 1801)