Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$

Question

I'm having some trouble figuring out how to give a direct proof using the definitions of mod m and congruence modulo m, without using any theorem involving mod or congruence.

if $a, b, c$ & $m$ are integers such that $m \geq 2$, $c > 0$ and $a \equiv b \pmod{m}$, then $ac \equiv bc\pmod{mc}$

What's your definition of congruence? In the common one it should follow from a simple multiplication. — Josselin Poiret, Nov 02 '16 at 22:12

score 2 · Accepted Answer · answered Nov 02 '16 at 22:12

2

$$a \equiv b \pmod{m}\iff b=a+km \implies cb=ca+kcm\implies ca \equiv cb \pmod{cm}.$$

answered Nov 02 '16 at 22:12

mfl

29,399

score 1 · Answer 2 · answered Nov 02 '16 at 22:23

1

Hint $\,\ m\mid n\, \Rightarrow\, mc\mid nc.\ $ OP has $\ n = a-b$

answered Nov 02 '16 at 22:23

Bill Dubuque

272,048

Thank you. Very helpful. – Avv Mar 18 '21 at 00:45

Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$

2 Answers2

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