Foundations — Divisibility rules — 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 (with proofs where possible)

Before you can prove a single divisibility rule, you must own a handful of tools. The parent note Divisibility Rules throws around symbols like $\equiv$, $(\mathrm{mod}m)$, $d_k$, $10^k$, and $\gcd$ as if you already knew them. This page builds each one from nothing, in the order you need them.

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1. What "divides" even means

Picture a row of $b$ dots. Try to fence them into groups of $a$. If the last fence lands exactly at the end, $a$ divides $b$. If a few lonely dots stick out past the last fence, those leftovers are the remainder.

The word remainder is the hero of this whole chapter. Hold onto it.

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2. Place value: a number is a stack of digits

When you write $1234$, the little symbols $1,2,3,4$ are called digits — each one is a single character from $0$ to $9$.

Why bother naming them? Because the position of a digit changes its value. The $2$ in $1234$ is not worth two — it is worth two hundred. That is the whole meaning of place value.

Read this out loud as a recipe: "take the far-left digit, multiply by its power of ten, add the next digit times its power of ten, ... down to the last digit times $10^0=1$." The $\dots$ (three dots) just means "keep the obvious pattern going".

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3. Powers of ten: $10^k$

One special case people forget: $10^0=1$. Anything to the power $0$ is $1$. That is why the last digit $d_0$ is multiplied by $1$ (it keeps its face value).

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4. The remainder, made precise: $(\mathrm{mod}m)$

Now the big one. When you divide $b$ by $m$, you get some remainder — a number from $0$ up to $m-1$. We give that idea a compact notation.

Examples: $17\mathrm{mod}5=2$ (because $17=3\times 5+2$). $20\mathrm{mod}5=0$ (nothing left over — so $5$ divides $20$).

Key link: $m$ divides $n$ exactly when $n\mathrm{mod}m=0$. "Zero remainder" and "divisible" are the same sentence.

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5. The congruence symbol: $\equiv$

The parent note writes things like $10\equiv 1(\mathrm{mod}3)$. That squiggly three-line equals sign is not ordinary equals.

So $10\equiv 1(\mathrm{mod}3)$ says: $10$ and $1$ have the same remainder when divided by $3$. Check: $10=3\times 3+1$ (remainder $1$), and $1=3\times 0+1$ (remainder $1$). Same remainder — congruent. ✓

The magic property (proved in Modular Arithmetic) is that congruences can be added and multiplied just like equations. If $10\equiv 1(\mathrm{mod}3)$, then $10^2\equiv 1^2=1(\mathrm{mod}3)$. That is why every power of ten collapses to $1$ in the rule for 3.

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6. Negative remainders: $10\equiv -1(\mathrm{mod}11)$

The rule for $11$ uses a surprising line: $10\equiv -1(\mathrm{mod}11)$. How can a remainder be negative?

Formally: $10\equiv -1(\mathrm{mod}11)$ because $11$ divides $10-(-1)=11$. ✓ This tidy $-1$ is what creates the alternating $+,-,+,-$ pattern in the rule for 11 — because $(-1{)}^0=+1$, $(-1{)}^1=-1$, $(-1{)}^2=+1$, and so on.

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7. gcd: the greatest common divisor

The rule for $6$ leans on the phrase $\gcd (2,3)=1$.

Why the topic needs it: to check divisibility by $6$, the note splits $6=2\times 3$ and checks $2$ and $3$ separately. This split is only legal because $\gcd (2,3)=1$. If the two factors shared a common factor, checking them separately would double-count and give wrong answers. This safety guarantee comes from the Chinese Remainder Theorem. Full details of computing gcd live in GCD and LCM.

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8. Putting it together: the prerequisite map

Here is how every foundation above flows into the divisibility rules:

Read it top to bottom: raw ideas (divides, digits, powers) feed the place-value recipe; the recipe plus congruence lets you reduce each power of ten mod $m$; and that single move branches into all the specific rules.

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9. A one-glance example of the whole machine

Notice: you never divided $1233$ by $3$. You only added its digits. That is the payoff of owning these symbols.

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Equipment checklist

Test yourself — say each answer out loud before revealing.