How is the expression "$a\equiv b \mod n$" pronounced?

Question

How is this expression pronounced ?

$$a\equiv b \mod n$$

For example:

$$24\equiv 4\mod 5$$

What is the "meaning" of mod? This word is a shortened version of which word? And what is the meaning of "modular"?

Answer 1

I would say something like "$a$ is equivalent [or congruent] to $b$ mod $n$"—"mod" here being pronounced to rhyme with "rod" or "pod" etc. Some people say "equal" instead of "equivalent"; as long as the "mod $n$" part is there, it doesn't much matter.

---

It just now occurred to me that you also asked what the term "modulus" or "modulo" means. Practically, if we say that

$$ a \equiv b \bmod n $$

we mean that $a$ and $b$ both leave the same remainder when divided by $n$; that is, there exist integer values $p$, $q$, and $r$ such that

$$ a = pn + r $$ $$ b = qn + r $$ $$ 0 \leq r < n $$

It's called "modular" arithmetic from Latin modulus "little measure"; the implication is that the modulus $n$ is what you use to measure out the quantities $a$ and $b$. What's left (if anything) is the same in either case.

Answer 2

$a$ is congruent to $b$ modulo $n$.

It means that they have the same remainder when you divide them by $n$, or shortly:

$$n\ \text{ divides }\ b-a$$

The "meaning", roughly, is that $a$ and $b$ become "equivalent" if you consider $n$ to be "equivalent" to $0$.

Just like $90$ and $-270$ are equivalent if you consider $360$ to be the same as $0$... if that rings a bell.

Answer 3

It is short for "modulo", and that's how it is pronounced, "$24$ equals $4$ modulo $5$". Colloquially, it means that 4 is the reminder of the división $24/4$. More exactly, that $4$ is the standard representative of the equivalence class of $24$ in the ring $\mathbb{Z}/5$.