Interactive factor tree

What a factor tree is

A factor tree breaks a number into factors, then breaks those factors down, until every branch ends in a prime. Reading the leaves gives the prime factorization: 36 = 2 × 2 × 3 × 3, or 2² × 3² with exponents. It is a tidy way to see what a number is made of.

On this page you pick a number from the buttons, type any number from 4 to 200, or press Surprise me. Tap a yellow circle and choose one of its factor pairs, such as 4 × 9 or 6 × 6; only pairs that really multiply to that number are offered. Prime numbers turn green and can't be split. When every leaf is green, the readout shows the prime factorization with exponents. Undo split takes back the last split.

Why build it by hand

The striking thing about factor trees is that different trees for the same number always end with the same primes. Start 36 with 4 × 9 or with 2 × 18 and you still finish with two 2s and two 3s. That fact, sometimes called the fundamental theorem of arithmetic, is easier to believe after building two different trees and seeing it happen. The tool notices when you finish a second tree with a different first split.

Common mix-ups

• Splitting a number into 1 × itself. That never gets you closer to primes, so the tool only offers pairs without 1.
• Stopping too soon at a number like 9 or 15. Only green leaves are prime; a yellow leaf still has factors.
• Thinking there is one right tree. Any first split works, and the primes at the end are the same.
• Writing 2³ as 2 × 3. The exponent says how many 2s are multiplied: 2³ = 2 × 2 × 2 = 8.

A 10-minute factor tree routine

• Pick 24. Ask the child which split they will choose and why, then build the tree.
• Read the leaves aloud and then the factorization with exponents: 2³ × 3.
• Press Try a different first split and build a new tree for 24. Compare the leaves.
• Pick 100 and ask the child to predict the factorization before splitting.
• Finish with a number of the child's choice and check it on paper by multiplying the primes back together.

Where it fits by grade

• Grade 4: factor pairs and prime or composite numbers up to 100 (4.OA.B.4).
• Grades 5 to 7: prime factorization is a common way to find greatest common factors and least common multiples, and to simplify fractions. Exponents arrive in grade 6 (6.EE.A.1).
• England: Year 5 introduces prime factors and composite numbers, and Year 6 uses common factors and multiples. Writing a number as a product of prime factors is part of Key Stage 3, from Year 7.

Learning more

Exponents and simplifying fractions are skills on Eurekami's skill map, and Daily Practice brings them back when they start to fade. When a factor question is tricky, the AI tutor asks guiding questions so the student does the reasoning.