Factors and Multiples • Topic 7 of 7

Factor Trees

A factor tree is a diagram for finding a prime factorization. Start with the number, split it into any factor pair, then keep splitting each composite branch until every leaf is prime. The product of all the leaves equals the original number, and the multiset of leaves is the same no matter which split you start with. For example, 36 → 6 × 6 → (2 × 3)(2 × 3) gives leaves 2, 2, 3, 3, so 36 = 2² × 3². Factor trees make the divisor-counting rule, GCF and LCM visual, and are an easy way to avoid arithmetic slips on the SAT.

✅ Solved examples

1. Use a factor tree to find the prime factorization of 48.
Split 48 → 6 × 8. Then 6 → 2 × 3 and 8 → 2 × 4 → 2 × 2 × 2. Leaves: 2, 3, 2, 2, 2 = 2⁴ × 3. So 48 = 2⁴ × 3.
2. The leaves of a factor tree are 2, 2, 3, 5. What number does the tree represent?
Multiply the leaves: 2 × 2 × 3 × 5 = 60. The number is 60 (= 2² × 3 × 5).
3. A factor tree has five leaves, all equal to 2. What is the number and how many divisors does it have?
Five 2s multiply to 2⁵ = 32. Number of divisors = exponent + 1 = 5 + 1 = 6.
4. Two factor trees give leaves {2, 2, 3} and {2, 3, 3}. What is the GCF of the two numbers?
The numbers are 2² × 3 = 12 and 2 × 3² = 18. GCF takes lowest shared powers: 2¹ × 3¹ = 6.

✏️ Practice — try these, take hints as needed

1. A factor tree has leaves 2, 3, 3. What number is it, and what is its prime factorization?
Multiply all the leaves together.
2 × 3 × 3.
Group the repeated prime as a power.
18 = 2 × 3².
2. Build a factor tree for 100 and list its prime leaves.
Split 100 → 10 × 10.
Split each 10 → 2 × 5.
Collect all four leaves.
Leaves 2, 2, 5, 5; so 100 = 2² × 5².
3. A factor tree for a number has leaves 2, 2, 2, 7. How many positive divisors does the number have?
Multiply the leaves to get the number, or read off exponents.
Leaves give 2³ × 7.
Divisors = (3+1)(1+1).
8 divisors (the number is 56).
4. The prime leaves of a factor tree are 3, 3, 3. What number and which power of 3 is it?
Three identical leaves of 3.
3 × 3 × 3.
Write as a power.
27 = 3³.
5. Two numbers have factor-tree leaves {2, 2, 5} and {2, 5, 5}. Find their LCM.
The numbers are 2² × 5 = 20 and 2 × 5² = 50.
LCM takes the highest power of each prime.
2² × 5² = 4 × 25.
100.

📝 Topic test — 8 questions

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