How to Factor Numbers

How to Factor Numbers

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A factor of a whole number is a whole number that divides it evenly. For example, \(6\) is a factor of \(24\) because \(24 \div 6 = 4\) with no remainder. The matching factor is \(4\), so factors usually come in pairs: \(6 \times 4 = 24\).

Every positive whole number has at least two factors: \(1\) and the number itself. A number with exactly two positive factors is called a prime number. A number with more than two positive factors is called a composite number.

Prime Factorization

A prime factorization writes a number as a product of prime numbers only. For example, \(8 = 2 \times 2 \times 2 = 2^3\).
prime_factors_of_8
How to Calculate Prime Factorization of 8?

How to Factor a Number

  • Try the smallest prime number first: \(2\). If the number is even, divide by \(2\).
  • If \(2\) no longer works, try \(3\), then \(5\), \(7\), \(11\), and so on.
  • Keep dividing until the quotient is \(1\). The prime numbers you divided by are the prime factors.

For example, \(84 \div 2 = 42\), \(42 \div 2 = 21\), \(21 \div 3 = 7\), and \(7 \div 7 = 1\). Therefore, \(84 = 2 \times 2 \times 3 \times 7\).

How to Factor Numbers Video

Factoring Numbers

Think of this lesson as more than a rule to memorize. Factoring Numbers is about number sense, equivalent forms, and careful arithmetic. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Fractions compare a part to a whole. Keep track of the numerator, denominator, and whether the pieces are the same size before adding, subtracting, or simplifying.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Look for a greatest common factor first.
  • Identify the pattern: trinomial, difference of squares, or grouping.
  • Choose factor pairs that rebuild the middle term.
  • Multiply the factors back to check.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Related Topics

How to Find the Greatest Common Factor
What are the Divisibility Rules
How to Find the Least Common Multiple
How to Simplify Fractions

Exercises for Factoring Numbers

1) Find the prime factorization of \(12\).

2) Find the prime factorization of \(18\).

3) Find the prime factorization of \(25\).

4) Find the prime factorization of \(36\).

5) Find the prime factorization of \(49\).

6) Find the prime factorization of \(54\).

7) Find the prime factorization of \(64\).

8) Find the prime factorization of \(75\).

9) Find the prime factorization of \(96\).

10) Find the prime factorization of \(121\).

11) Find the prime factorization of \(144\).

12) Find the prime factorization of \(168\).

13) Find the prime factorization of \(210\).

14) Find the prime factorization of \(225\).

15) Find the prime factorization of \(256\).

16) Find the prime factorization of \(315\).

17) Find the prime factorization of \(420\).

18) Find the prime factorization of \(512\).

19) Find the prime factorization of \(693\).

20) Find the prime factorization of \(840\).

1) \(12 = \color{red}{2 \times 2 \times 3}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(12 \div 2 = 6\)
\(6 \div 2 = 3\)
\(3 \div 3 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(12 = \color{red}{2 \times 2 \times 3}\).
2) \(18 = \color{red}{2 \times 3 \times 3}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(18 \div 2 = 9\)
\(9 \div 3 = 3\)
\(3 \div 3 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(18 = \color{red}{2 \times 3 \times 3}\).
3) \(25 = \color{red}{5 \times 5}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(25 \div 5 = 5\)
\(5 \div 5 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(25 = \color{red}{5 \times 5}\).
4) \(36 = \color{red}{2 \times 2 \times 3 \times 3}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(36 \div 2 = 18\)
\(18 \div 2 = 9\)
\(9 \div 3 = 3\)
\(3 \div 3 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(36 = \color{red}{2 \times 2 \times 3 \times 3}\).
5) \(49 = \color{red}{7 \times 7}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(49 \div 7 = 7\)
\(7 \div 7 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(49 = \color{red}{7 \times 7}\).
6) \(54 = \color{red}{2 \times 3 \times 3 \times 3}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(54 \div 2 = 27\)
\(27 \div 3 = 9\)
\(9 \div 3 = 3\)
\(3 \div 3 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(54 = \color{red}{2 \times 3 \times 3 \times 3}\).
7) \(64 = \color{red}{2 \times 2 \times 2 \times 2 \times 2 \times 2}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(64 \div 2 = 32\)
\(32 \div 2 = 16\)
\(16 \div 2 = 8\)
\(8 \div 2 = 4\)
\(4 \div 2 = 2\)
\(2 \div 2 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(64 = \color{red}{2 \times 2 \times 2 \times 2 \times 2 \times 2}\).
8) \(75 = \color{red}{3 \times 5 \times 5}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(75 \div 3 = 25\)
\(25 \div 5 = 5\)
\(5 \div 5 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(75 = \color{red}{3 \times 5 \times 5}\).
9) \(96 = \color{red}{2 \times 2 \times 2 \times 2 \times 2 \times 3}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(96 \div 2 = 48\)
\(48 \div 2 = 24\)
\(24 \div 2 = 12\)
\(12 \div 2 = 6\)
\(6 \div 2 = 3\)
\(3 \div 3 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(96 = \color{red}{2 \times 2 \times 2 \times 2 \times 2 \times 3}\).
10) \(121 = \color{red}{11 \times 11}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(121 \div 11 = 11\)
\(11 \div 11 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(121 = \color{red}{11 \times 11}\).
11) \(144 = \color{red}{2 \times 2 \times 2 \times 2 \times 3 \times 3}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(144 \div 2 = 72\)
\(72 \div 2 = 36\)
\(36 \div 2 = 18\)
\(18 \div 2 = 9\)
\(9 \div 3 = 3\)
\(3 \div 3 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(144 = \color{red}{2 \times 2 \times 2 \times 2 \times 3 \times 3}\).
12) \(168 = \color{red}{2 \times 2 \times 2 \times 3 \times 7}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(168 \div 2 = 84\)
\(84 \div 2 = 42\)
\(42 \div 2 = 21\)
\(21 \div 3 = 7\)
\(7 \div 7 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(168 = \color{red}{2 \times 2 \times 2 \times 3 \times 7}\).
13) \(210 = \color{red}{2 \times 3 \times 5 \times 7}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(210 \div 2 = 105\)
\(105 \div 3 = 35\)
\(35 \div 5 = 7\)
\(7 \div 7 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(210 = \color{red}{2 \times 3 \times 5 \times 7}\).
14) \(225 = \color{red}{3 \times 3 \times 5 \times 5}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(225 \div 3 = 75\)
\(75 \div 3 = 25\)
\(25 \div 5 = 5\)
\(5 \div 5 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(225 = \color{red}{3 \times 3 \times 5 \times 5}\).
15) \(256 = \color{red}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(256 \div 2 = 128\)
\(128 \div 2 = 64\)
\(64 \div 2 = 32\)
\(32 \div 2 = 16\)
\(16 \div 2 = 8\)
\(8 \div 2 = 4\)
\(4 \div 2 = 2\)
\(2 \div 2 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(256 = \color{red}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}\).
16) \(315 = \color{red}{3 \times 3 \times 5 \times 7}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(315 \div 3 = 105\)
\(105 \div 3 = 35\)
\(35 \div 5 = 7\)
\(7 \div 7 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(315 = \color{red}{3 \times 3 \times 5 \times 7}\).
17) \(420 = \color{red}{2 \times 2 \times 3 \times 5 \times 7}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(420 \div 2 = 210\)
\(210 \div 2 = 105\)
\(105 \div 3 = 35\)
\(35 \div 5 = 7\)
\(7 \div 7 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(420 = \color{red}{2 \times 2 \times 3 \times 5 \times 7}\).
18) \(512 = \color{red}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(512 \div 2 = 256\)
\(256 \div 2 = 128\)
\(128 \div 2 = 64\)
\(64 \div 2 = 32\)
\(32 \div 2 = 16\)
\(16 \div 2 = 8\)
\(8 \div 2 = 4\)
\(4 \div 2 = 2\)
\(2 \div 2 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(512 = \color{red}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}\).
19) \(693 = \color{red}{3 \times 3 \times 7 \times 11}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(693 \div 3 = 231\)
\(231 \div 3 = 77\)
\(77 \div 7 = 11\)
\(11 \div 11 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(693 = \color{red}{3 \times 3 \times 7 \times 11}\).
20) \(840 = \color{red}{2 \times 2 \times 2 \times 3 \times 5 \times 7}\)
Solution:
Step 1: Start with the smallest prime divisor and keep dividing by primes until the quotient is \(1\).
\(840 \div 2 = 420\)
\(420 \div 2 = 210\)
\(210 \div 2 = 105\)
\(105 \div 3 = 35\)
\(35 \div 5 = 7\)
\(7 \div 7 = 1\)
Step 2: The divisors used are all prime, so the prime factorization is \(840 = \color{red}{2 \times 2 \times 2 \times 3 \times 5 \times 7}\).

Factor Numbers Quiz