Factors & Multiples

A factor of a number divides into it evenly, leaving no remainder. If $a \times b = n$, then both $a$ and $b$ are factors of $n$.

To find all the factors of a number, work in pairs. Start at $1$ and count upward: for each number that divides evenly, write down both partners. Stop once the pairs start to repeat.

A multiple of a number is what you get by multiplying it by $1, 2, 3, 4, \dots$ In other words, it is any number in that number's “times table.” Multiples go on forever, while factors are a short, finite list.

• A prime number has exactly two factors: $1$ and itself. The first primes are $2, 3, 5, 7, 11, 13, 17, 19, 23, \dots$
• A composite number has more than two factors --- it can be broken into smaller factors.
• The number $1$ is neither prime nor composite: it has only one factor.
• $2$ is the only even prime; every other even number has $2$ as an extra factor.

These shortcuts tell you if a number divides evenly without doing the long division.

• 2: the last digit is even ($0, 2, 4, 6, 8$).
• 3: the sum of the digits is divisible by $3$.
• 4: the number formed by the last two digits is divisible by $4$.
• 5: the last digit is $0$ or $5$.
• 6: it passes the tests for both $2$ and $3$.
• 9: the sum of the digits is divisible by $9$.
• 10: the last digit is $0$.

The prime factorization of a number writes it as a product of prime numbers only. Every composite number has exactly one prime factorization (in some order). A factor tree is the easiest way to find it: split the number into any two factors, then keep splitting until every branch ends in a prime. Finally, write repeated primes using exponents.

The greatest common factor of two numbers is the biggest number that divides both. There are two reliable methods:

• Listing: list the factors of each number, then pick the largest one they share.
• Prime factorization: factor each number into primes, then multiply the primes they have in common (using the smaller exponent of each shared prime).

The least common multiple of two numbers is the smallest number that both divide into --- the first multiple they share. Again, two methods:

• Listing: list the multiples of each number until one appears in both lists.
• Prime factorization: factor each number, then multiply every prime that appears in either number, using the larger exponent of each.

The hardest part of a word problem is choosing the right tool. Watch for these signals:

• Use the GCF when you split or group things into the largest equal piles, rows, or teams --- the numbers get smaller. Clues: “greatest,” “largest,” “equal groups,” “split evenly.”
• Use the LCM when events repeat and you want the next time they line up, or the smallest amount that works for both --- the numbers get bigger. Clues: “at the same time,” “again,” “least,” “smallest,” “every.”

Once a number is written in prime-power form, you can count how many factors it has without listing a single one. If

then the number of divisors is

The idea: when building a divisor, the prime $p$ can appear $0, 1, 2, \dots,$ or $a$ times --- that is $a+1$ independent choices. Multiplying the choices for each prime counts every divisor exactly once.

You can also add up all the divisors straight from the prime factorization. For each prime power, sum the powers from $p^0$ up to $p^a$, then multiply those sums together:

Multiplying the parentheses out reproduces every divisor of $n$ exactly once, so their grand total falls out automatically.

For any two whole numbers $a$ and $b$, the greatest common factor and least common multiple are locked together by

This makes finding the LCM effortless once you know the GCF (or vice versa): just rearrange.

Why it works: for each prime, the GCF takes the smaller exponent and the LCM takes the larger one --- together they use up exactly the two exponents that appear in $a$ and $b$.

For large numbers, factoring can be slow. The Euclidean algorithm finds the GCF using only repeated division with remainder. Repeatedly replace the larger number by the remainder when it is divided by the smaller:

and stop when the remainder hits $0$. The last nonzero remainder is the GCF. It works because any common factor of $a$ and $b$ is also a factor of their remainder.

Compare a number to the sum of its proper divisors (all its divisors except itself):

• Perfect: the proper divisors add up to the number exactly. The smallest is $6 = 1 + 2 + 3$; the next is $28 = 1 + 2 + 4 + 7 + 14$.
• Abundant: the proper divisors sum to more than the number. Example: $12$, since $1 + 2 + 3 + 4 + 6 = 16 > 12$.
• Deficient: the proper divisors sum to less than the number. Example: $10$, since $1 + 2 + 5 = 8 < 10$. Every prime is deficient (its only proper divisor is $1$).