Factor Tree for 36

Find the prime factorization of 36.

Start with 36 and find any factor pair: $36 = 6 \times 6$ (or $4 \times 9$, $2 \times 18$, etc.) = Choose $6 \times 6$

Factor each 6: $6 = 2 \times 3$ (both are prime) = Circle 2 and 3

Collect all prime factors: $36 = 2 \times 3 \times 2 \times 3$ = Four prime factors

Write in exponential form: Two 2s and two 3s: $2^2 \times 3^2$ = $36 = 2^2 \times 3^2$

Answer: $36 = 2^2 \times 3^2$

Factor Tree for 60

Find the prime factorization of 60.

Start with an easy factor: $60 = 2 \times 30$ (60 is even) = Circle 2 (prime)

Factor 30: $30 = 2 \times 15$ (30 is even) = Circle 2 (prime)

Factor 15: $15 = 3 \times 5$ (both prime) = Circle 3 and 5

Write the factorization: $60 = 2 \times 2 \times 3 \times 5$ = $60 = 2^2 \times 3 \times 5$

Answer: $60 = 2^2 \times 3 \times 5$

Factor Tree for 84

Find the prime factorization of 84.

Divide by 2 (even number): $84 = 2 \times 42$ = Circle 2

Factor 42: $42 = 2 \times 21$ = Circle 2

Factor 21: $21 = 3 \times 7$ (both prime) = Circle 3 and 7

Combine and simplify: $84 = 2 \times 2 \times 3 \times 7$ = $84 = 2^2 \times 3 \times 7$

Answer: $84 = 2^2 \times 3 \times 7$

Factor Tree for 180

Find the prime factorization of 180.

Start by dividing by 10: $180 = 10 \times 18$ = Split into two branches

Factor 10: $10 = 2 \times 5$ (both prime) = Circle 2 and 5

Factor 18: $18 = 2 \times 9$, then $9 = 3 \times 3$ = Circle 2, 3, 3

Collect all primes: $180 = 2 \times 5 \times 2 \times 3 \times 3$ = $180 = 2^2 \times 3^2 \times 5$

Answer: $180 = 2^2 \times 3^2 \times 5$

Mistake: Stopping before all factors are prime

Why: Students sometimes stop at composite numbers like 4 or 9, thinking they are done.

Correct: Keep factoring until every number at the bottom of your tree is prime (2, 3, 5, 7, 11...).

Mistake: Forgetting to include repeated factors

Why: When 36 = 6 x 6, students might write 2 x 3 instead of 2 x 2 x 3 x 3.

Correct: Count ALL prime factors from EVERY branch of the tree.

Mistake: Thinking 1 is a prime number

Why: 1 divides every number, but it is not considered prime.

Correct: Prime numbers start at 2. The number 1 is neither prime nor composite.

Mistake: Getting different answers with different factor trees

Why: Students worry when they start with different factors (2 x 18 vs 6 x 6).

Correct: The final prime factorization is always the same, no matter which factor pair you start with!

Internet Security (RSA Encryption)

Online banking and secure websites use encryption based on multiplying two very large prime numbers. Finding the prime factors of this product is extremely difficult, which keeps your data safe.

A 2048-bit encryption key uses primes with over 300 digits. Even the fastest computers would take millions of years to factor it!

Music and Rhythm

Musicians use prime factorization to understand time signatures and rhythmic patterns. Finding common factors helps in creating polyrhythms.

A rhythm in 12 beats ($2^2 \times 3$) can be divided into groups of 2, 3, 4, or 6.

Q: Does it matter which factor pair I start with?

A: No! You can start with any factor pair and you will always get the same prime factorization. For example, 36 = 2 x 18 or 36 = 4 x 9 or 36 = 6 x 6 all lead to $2^2 \times 3^2$.

Q: What if the number is already prime?

A: Then you're done! The prime factorization of a prime number (like 17) is just itself: $17 = 17$.

Q: Why isn't 1 considered a prime number?

A: If 1 were prime, then every number would have infinitely many prime factorizations (36 = 2 x 2 x 3 x 3 = 1 x 2 x 2 x 3 x 3 = 1 x 1 x 2 x ...). To keep the Fundamental Theorem true, we define primes as starting at 2.