What is a factor of a number?

A factor is a whole number that divides another number with no remainder. For example, 1, 2, 3, 4, 6 and 12 are all factors of 12, because each one divides 12 evenly.

What is the difference between factors and prime factors?

Factors are all the numbers that divide evenly into a number. Prime factors are only the factors that are prime. The factors of 12 are 1, 2, 3, 4, 6 and 12, but its prime factors are just 2 and 3.

What is the difference between factors and multiples?

Factors divide into a number, so they are always equal to or smaller than it. Multiples are what you get by multiplying a number, so they are equal to or larger than it. 3 is a factor of 12, and 12 is a multiple of 3.

No. A prime has exactly two distinct factors, 1 and itself, and 1 has only one factor. Mathematicians also exclude 1 so that prime factorization stays unique. 1 is neither prime nor composite.

Why do perfect squares have an odd number of factors?

Factors normally come in pairs, like 2 × 18 for 36. In a perfect square, one pair repeats the same number (6 × 6), so that factor is counted once. This leaves an odd total: 36 has 9 factors.

Do numbers have negative factors?

Yes. If 3 × 4 = 12, then (−3) × (−4) = 12 as well, so −3 and −4 are also factors of 12. Most school problems ask only for positive factors. Switch on the ± button in the calculator to include the negatives.

How are GCF and LCM related?

For two numbers a and b, GCF(a, b) × LCM(a, b) = a × b. If you know the GCF, you can find the LCM by dividing the product of the numbers by the GCF. The GCF · LCM mode shows this check for you.

How large a number can the calculator handle?

Any whole number from 1 up to 1,000,000,000,000,000 (one quadrillion). Everything runs in your browser, so your numbers are never sent anywhere.

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What Are Factors of a Number?

A factor of a number is a whole number that divides it exactly, with nothing left over. Since 12 ÷ 4 = 3, both 4 and 3 are factors of 12. Factors always come in pairs, because every division that works gives you two numbers: the divisor and the answer.

Factors are the building blocks of arithmetic. You use them to simplify fractions, find common denominators, share things fairly and factor algebra. A few terms are worth keeping straight:

Prime number: greater than 1, and its only factors are 1 and itself, like 2, 3, 5, 7 and 11.
Composite number: has more than two factors, like 12 with 1, 2, 3, 4, 6 and 12.
Multiple: what you get by multiplying a number, so 12 is a multiple of 4 while 4 is a factor of 12.

How to Find All the Factors of a Number

The fastest way to list every factor is to work in pairs. Start with 1 × n, then test 2, 3, 4 and so on. Whenever a division is exact, write down both numbers. Take 36:

Handwritten notebook page listing the factor pairs of 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6, with the factors 1, 2, 3, 4, 6, 9, 12, 18, 36 circled in yellow highlighter — Factors of 36, written as pairs and then listed in order.

You can stop at the square root. Past √n you only meet pairs you already wrote down, so for 36 you stop at 6. Because 6 × 6 = 36 pairs with itself, perfect squares always have an odd number of factors. 36 has nine.

Quick steps

Write 1 × n as your first pair.
Test 2, 3, 4… in order, and record both numbers of each exact division.
Stop when you reach the square root of n.
List every number from the pairs from smallest to largest.

Prime Factorization with a Factor Tree

Prime factorization writes a number as a product of primes only. A factor tree is the most visual way to do it: pick any two numbers that multiply to your number, write them as branches, and keep splitting every branch that is not prime. When every branch ends in a prime, multiply the primes together.

Handwritten factor tree of 60 splitting into 2 and 30, then 30 into 2 and 15, then 15 into 3 and 5, with the result 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5 highlighted — Factor tree of 60. Circled numbers are primes, so those branches stop.

So 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. It does not matter which pair you start with. This is the Fundamental Theorem of Arithmetic: every whole number above 1 is a product of primes in exactly one way.

Prime Factorization by Repeated Division

If trees feel messy, use the ladder method. Divide by the smallest prime that fits, write the quotient underneath, and repeat until you reach 1. The primes down the left side are your answer.

Handwritten division ladder for 84: divide by 2 to get 42, by 2 to get 21, by 3 to get 7, by 7 to get 1, giving 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7 — The ladder method for 84: 84 = 2² × 3 × 7.

There is a bonus. Add 1 to each exponent and multiply: (2 + 1) × (1 + 1) × (1 + 1) = 12. That is the total number of factors of 84, found without listing a single one.

How to Find the GCF and LCM of Two Numbers

The greatest common factor (GCF) is the largest number that divides both numbers. The least common multiple (LCM) is the smallest number both divide into. Prime factorizations make both easy: for the GCF take the lowest power of each shared prime, and for the LCM take the highest power of every prime.

Handwritten work finding the GCF and LCM of 24 and 36 using prime factorizations: GCF = 12 and LCM = 72, checked by 12 × 72 = 864 = 24 × 36 — GCF and LCM of 24 and 36, with the product check at the bottom.

With 24 = 2³ × 3 and 36 = 2² × 3², the GCF is 2² × 3 = 12 and the LCM is 2³ × 3² = 72. A handy check is that GCF × LCM = a × b, and here 12 × 72 = 864 = 24 × 36.

How to Tell if a Number Is Prime or Composite

To test whether n is prime, divide it by every prime up to √n. If any division is exact, n is composite. If none are, n is prime. Always test 7 before you decide, because numbers like 91 only look prime.

Handwritten test showing that 91 is not prime: the square root of 91 is about 9.5, 2, 3 and 5 do not divide it, but 91 ÷ 7 = 13, so 91 = 7 × 13 is composite — 91 passes the 2, 3 and 5 tests but fails at 7, so it is composite.

Divisibility Rules That Make Factoring Faster

You can often tell whether a small number is a factor without dividing. A number is divisible by:

2 if the last digit is even (0, 2, 4, 6 or 8). 138 ends in 8, so it is divisible by 2.
3 if the sum of the digits is divisible by 3. 4,521 gives 4 + 5 + 2 + 1 = 12, so yes.
4 if the last two digits form a number divisible by 4. 1,316 ends in 16, and 16 ÷ 4 = 4, so yes.
5 if the last digit is 0 or 5. 2,785 ends in 5, so yes.
6 if the number is divisible by both 2 and 3. 324 is even and its digits add to 9, so yes.
8 if the last three digits form a number divisible by 8. 5,432 ends in 432, and 432 ÷ 8 = 54, so yes.
9 if the sum of the digits is divisible by 9. 7,299 gives 7 + 2 + 9 + 9 = 27, so yes.
10 if the last digit is 0. 9,370 ends in 0, so yes.

Use these shortcuts before every method above, then type your number into the calculator to confirm the result.

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