What does the squiggly symbol \equiv mean?

That is the congruence symbol. When you write n \equiv 0 \pmod{d}, it means n and 0 have the same remainder when divided by d — in other words, n is divisible by d. It is not an equals sign, but for this purpose it means the same thing.

Why do I need to learn this instead of just doing regular division?

Divisibility rules are a foundation for many topics later: when fractions simplify, when numbers are prime, and how patterns work with remainders. The modulo notation is also much faster to write and reason about than saying "divide and check the remainder."

Can 0 be divisible by a number, like is 0 \equiv 0 \pmod{5}?

Yes — 0 is divisible by every nonzero number because 0 \div 5 = 0 with remainder 0. So 0 \equiv 0 \pmod{d} is always true for any divisor d.

What if the remainder is not 0? Can I still write a modulo equation?

Yes. If n is not divisible by d, you write the actual remainder. For example, 17 \div 5 = 3 R 2, so 17 \equiv 2 \pmod{5}. The remainder just will not be 0.

Divisibility Rules

Divisibility rules determine when one number divides into another evenly with no remainder — use them to quickly check if a number is a multiple.

n \equiv 0 \pmod{d}

Solve a problem with Divisibility Rules

Type the problem. The solver will use Divisibility Rules where Divisibility Rules is the right tool, and tell you when it is not.

What each symbol means

What Divisibility Rules takes

Symbol	Meaning
$n$	The number you are dividing — the larger number in a division problem.
$d$	The divisor — the number you're dividing by, or the number you're checking to see if it goes into $n$ evenly.

When to use it

When you need to know if one number divides evenly into another without doing full long division.

Level

Usually taught in: Pre-Algebra

Worked examples

1. Check if 15 is divisible by 3

Problem

Determine whether

15

is divisible by

3

— that is, whether

15 \equiv 0 \pmod{3}

$15 \div 3 = 5$
Divide $n = 15$ by $d = 3$ to see what happens.
$5 \times 3 = 15$
Check: multiply the answer back. We get exactly $15$ , not $15$ plus something left over.
$15 \text{ has remainder } 0$
Since we get exactly $15$ with no leftover, the remainder is $0$ .

Answer: $15 \equiv 0 \pmod{3} \text{ — yes, 15 is divisible by 3}$

When we divide $15$ by $3$ , we get a whole number ( $5$ ) with nothing left over. That remainder of $0$ is the key: it means $3$ divides into $15$ evenly, so $15$ is divisible by $3$ .

2. Check if 37 is divisible by 5

Problem

Determine whether

37

is divisible by

5

— that is, whether

37 \equiv 0 \pmod{5}

$37 \div 5 = 7 \text{ R } 2$
Divide $n = 37$ by $d = 5$ . The quotient is $7$ and the remainder is $2$ .
$7 \times 5 = 35$
Multiply back: $5$ goes into $37$ a total of $7$ times, making $35$ .
$37 - 35 = 2$
Subtract to find the remainder: $37 - 35 = 2$ . This leftover is the remainder.

Answer: $37 \equiv 2 \pmod{5} \text{ — no, 37 is not divisible by 5}$

Since the remainder is $2$ (not $0$ ), $5$ does not divide evenly into $37$ . The symbol $37 \equiv 2 \pmod{5}$ means the remainder is $2$ , so $37$ is not divisible by $5$ .

3. Decide if a gift can be split equally

Problem

A teacher has

28

pencils to give equally to

4

students. Can each student get the same whole number of pencils with none left over? Check whether

28

is divisible by

4

$28 \div 4 = 7$
Divide the total ( $28$ ) by the number of students ( $4$ ). We get $7$ with no remainder.
$7 \times 4 = 28$
To verify: if each student gets $7$ pencils, then $7 \times 4 = 28$ pencils total—exactly what we have.
$28 \equiv 0 \pmod{4}$
Since the remainder is $0$ , we can write this using modular notation: $28$ is congruent to $0$ modulo $4$ .

Answer: $\text{Yes, each student gets } 7 \text{ pencils with none left over}$

Because $28$ is divisible by $4$ (remainder is $0$ ), we can split the pencils equally. In modular arithmetic, $28 \equiv 0 \pmod{4}$ tells us exactly this: there is no remainder.

Common mistakes

Where Divisibility Rules usually goes wrong

The mistake	Why it is wrong	The fix
Writing $24 \equiv 0 \pmod{5}$ because the numbers look like they might go together.	Actually, $24 \div 5 = 4$ R $4$ , so the remainder is $4$ , not $0$ . You must actually do the division, not guess.	Divide: $24 \div 5 = 4$ R $4$ , so $24 \equiv 4 \pmod{5}$ — $24$ is not divisible by $5$ .
Confusing the quotient with the remainder, writing $15 \equiv 5 \pmod{3}$ because $15 \div 3 = 5$ .	The number in the modulo statement is the remainder, not the answer to the division. When $15 \div 3 = 5$ , the remainder is $0$ , not $5$ .	Write $15 \equiv 0 \pmod{3}$ — the remainder is $0$ because $3$ divides $15$ perfectly.
Claiming $-12 \not\equiv 0 \pmod{4}$ because the number is negative.	Divisibility works for negative numbers too: $-12 \div 4 = -3$ with remainder $0$ , so $-12$ is divisible by $4$ .	Check: $-12 \div 4 = -3$ R $0$ , so $-12 \equiv 0 \pmod{4}$ — negative numbers can be divisible.

Tips and when to use something else

When the remainder is $0$ , the number $n$ is divisible by $d$ — write it as $n \equiv 0 \pmod{d}$ .
Use the Modulo Operation if you need to understand what the remainder means symbolically; divisibility rules are just the $\equiv 0$ case.
For quick checks without long division, learn divisibility tests for $2, 3, 5, 9, 10$ — they use patterns in digits instead of dividing.

Related formulas

Modulo OperationThe modulo operation finds the remainder when one integer is divided by another, useful when tracking cycles and repetition patterns.Long DivisionLong Division breaks division into steps, giving you a whole-number quotient and a remainder, showing what divides evenly and what's left.Order of OperationsOrder of Operations is the set of rules that tells you which calculation to do first when you have multiple operations in one expression.Greatest Common FactorThe Greatest Common Factor finds the largest whole number that divides evenly into two or more numbers, helping you simplify fractions.

Frequently asked questions

What does the squiggly symbol $\equiv$ mean?: That is the congruence symbol. When you write $n \equiv 0 \pmod{d}$ , it means $n$ and $0$ have the same remainder when divided by $d$ — in other words, $n$ is divisible by $d$ . It is not an equals sign, but for this purpose it means the same thing.
Why do I need to learn this instead of just doing regular division?: Divisibility rules are a foundation for many topics later: when fractions simplify, when numbers are prime, and how patterns work with remainders. The modulo notation is also much faster to write and reason about than saying "divide and check the remainder."
Can $0$ be divisible by a number, like is $0 \equiv 0 \pmod{5}$ ?: Yes — $0$ is divisible by every nonzero number because $0 \div 5 = 0$ with remainder $0$ . So $0 \equiv 0 \pmod{d}$ is always true for any divisor $d$ .
What if the remainder is not $0$ ? Can I still write a modulo equation?: Yes. If $n$ is not divisible by $d$ , you write the actual remainder. For example, $17 \div 5 = 3$ R $2$ , so $17 \equiv 2 \pmod{5}$ . The remainder just will not be $0$ .

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Reviewed 2026-09-18