What is the difference between a divisor and a multiple?

A divisor (or factor) of an integer b is a nonzero integer a that divides b exactly, meaning b = a times k for some integer k. A multiple is the reverse relationship: b is a multiple of a. For example, 3 is a divisor of 12, and 12 is a multiple of 3.

What is the difference between a prime and composite number?

A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. A composite number is an integer greater than 1 that has at least one additional positive divisor. The number 1 is neither prime nor composite. Every composite number can be expressed as a unique product of primes.

What is the difference between GCD and LCM?

The greatest common divisor (GCD) of two integers is the largest positive integer that divides both. The least common multiple (LCM) is the smallest positive integer that both divide into evenly. They are related by the identity GCD(a, b) times LCM(a, b) equals the absolute value of a times b.

What is the difference between proper and improper fractions?

A proper fraction has a numerator smaller than the denominator, so its value is between 0 and 1. An improper fraction has a numerator equal to or greater than the denominator, so its value is 1 or more. Every improper fraction can be rewritten as a mixed number combining a whole part and a proper fraction.

What does congruence mean in modular arithmetic?

Two integers a and b are congruent modulo m if their difference is divisible by m, written a is congruent to b mod m. Equivalently, they have the same remainder when divided by m. The modulus m partitions all integers into exactly m residue classes based on their remainders.

Arithmetic Terms and Definitions

About This Glossary

This glossary organizes 22 arithmetic terms into three categories covering the core vocabulary of number theory and fractions.

Divisibility establishes the language of factors and multiples across 8 entries: divisor, multiple, prime number, composite number, prime factorization, coprime integers, greatest common divisor, and least common multiple. These terms describe how integers relate through division and form the foundation for working with fractions and modular arithmetic.

Fractions covers 11 entries on rational number representation: fraction, numerator, denominator, proper and improper fractions, mixed numbers, equivalent fractions, reciprocals, common denominators, and complex fractions. Each term addresses how parts of a whole are expressed, compared, and manipulated.

Modular Arithmetic addresses 4 entries on cyclic number systems: modulus, congruence, remainder, and residue class. These terms define how integers are grouped by their remainders and how arithmetic operates within those groups.

Each definition includes an intuitive explanation, key properties, examples, and links to the detailed lesson page. Use the search bar or category filters above to navigate.

DivisibilityFractionsModular Arithmetic

Divisibility(8)

Fractions(10)

Modular Arithmetic(4)

22 of 22 terms

22 terms

Divisibility

(8 items)

Divisor (Factor)

An integer

a \neq 0

is a divisor of an integer

b

if there exists an integer

k

such that

b = a \cdot k

, written

a \mid b

Factors

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Multiple

An integer

b

is a multiple of an integer

a \neq 0

if there exists an integer

k

such that

b = a \cdot k

Multiples

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Prime Number

An integer

p > 1

whose only positive divisors are

1

and

p

itself

Prime Numbers

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Composite Number

An integer

n > 1

that has at least one positive divisor other than

1

and

n

itself

Prime Numbers

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Prime Factorization

The expression of a positive integer

n > 1

as a product of prime powers:

n = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k}

, where each

p_i

is prime and each

a_i \geq 1

Prime Factorization

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Coprime (Relatively Prime)

Two integers

a

and

b

are coprime if their greatest common divisor is

1

\gcd(a, b) = 1

Coprime Numbers

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Greatest Common Divisor (GCD)

The largest positive integer

d

that divides both

a

and

b

d = \gcd(a, b)

where

d \mid a

and

d \mid b

, and for every common divisor

c

a

and

b

c \mid d

GCD

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Least Common Multiple (LCM)

The smallest positive integer

m

that is a multiple of both

a

and

b

m = \operatorname{lcm}(a, b)

where

a \mid m

and

b \mid m

, and for every common multiple

n

a

and

b

m \mid n

LCM

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Fractions

(10 items)

Fraction

An expression of the form

\frac{a}{b}

where

a

is the numerator and

b \neq 0

is the denominator, representing the quotient of

a

divided by

b

Fractions

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Numerator

In the fraction

\frac{a}{b}

, the integer

a

is the numerator — the number above the fraction bar

Numerator and Denominator

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Denominator

In the fraction

\frac{a}{b}

, the integer

b \neq 0

is the denominator — the number below the fraction bar

Numerator and Denominator

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Proper Fraction

A fraction

\frac{a}{b}

with

0 < a < b

, representing a value strictly between

0

and

1

Proper and Improper Fractions

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Improper Fraction

A fraction

\frac{a}{b}

with

a \geq b > 0

, representing a value greater than or equal to

1

Proper and Improper Fractions

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Mixed Number

A number written as

q\frac{r}{b}

, combining a whole number

q

and a proper fraction

\frac{r}{b}

, equivalent to

\frac{qb + r}{b}

Mixed Numbers

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Equivalent Fractions

Two fractions

\frac{a}{b}

and

\frac{c}{d}

are equivalent if

a \cdot d = b \cdot c

, written

\frac{a}{b} = \frac{c}{d}

Equivalent Fractions

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Reciprocal

The reciprocal of a nonzero number

\frac{a}{b}

\frac{b}{a}

, such that

\frac{a}{b} \cdot \frac{b}{a} = 1

Reciprocals

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Common Denominator

A common denominator of two fractions

\frac{a}{b}

and

\frac{c}{d}

is any positive integer

m

such that

b \mid m

and

d \mid m

. The least common denominator (LCD) is

\operatorname{lcm}(b, d)

Finding a Common Denominator

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Complex Fraction

A fraction in which the numerator, the denominator, or both contain a fraction:

\frac{\;\frac{a}{b}\;}{\frac{c}{d}}

Complex Fractions

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Modular Arithmetic

(4 items)

Modulus

A fixed positive integer

m

that defines the system of modular arithmetic, where integers are considered equivalent if they differ by a multiple of

m

Modular Arithmetic

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Congruence

Two integers

a

and

b

are congruent modulo

m

m \mid (a - b)

, written

a \equiv b \pmod{m}

Congruence

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Remainder

For integers

a

and

m > 0

, the remainder

r

is the unique integer satisfying

a = qm + r

with

0 \leq r < m

Modular Arithmetic

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Residue Class

The residue class of an integer

a

modulo

m

is the set

[a]_m = \{a + km : k \in \mathbb{Z}\}

— all integers congruent to

a

modulo

m

Properties of Congruence

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