Name
- Name the prime factors of 18.
- Write the prime factorization of 42 using multiplication.
Why
- Why is 1 not included in the prime factorization of a number?
- Why is (24 = 4 \times 6) not a prime factorization, but (24 = 2 \times 2 \times 2 \times 3) is?
Explain
- Explain how you could use a factor tree to find the prime factorization of 36.
- Explain how you know when a factor tree is complete.
Suggest
- Suggest a factor pair you could use to start finding the prime factorization of 90. Explain why it is a useful choice.
- Suggest a way to check that (2^2 \times 3 \times 5) is the prime factorization of 60.
Imagine
- Imagine a number has the prime factorization (2^2 \times 3 \times 5). What might its factor tree look like? Describe one possible tree.
- Imagine you are given a composite number but cannot use a factor tree. What strategy could you use to find its prime factorization?
Share
- Share how you would explain the difference between a factor and a prime factor to a classmate.
- Share one mistake someone might make when writing a prime factorization, and how you would help them correct it.
Answer key
Name
- The prime factors of 18 are 2 and 3. Its prime factorization is (2 \times 3 \times 3).
- (42 = 2 \times 3 \times 7).
Why
- 1 is not included because it is not prime. A prime number has exactly two positive factors: 1 and itself.
- A prime factorization uses only prime factors. Since 4 and 6 are composite, (4 \times 6) is not a prime factorization. The factors 2 and 3 are all prime, so (2 \times 2 \times 2 \times 3) is a prime factorization of 24.
Explain
- One possible factor tree starts with (36 = 6 \times 6), then splits each 6 into (2 \times 3). The prime factorization is (2 \times 2 \times 3 \times 3).
- A factor tree is complete when every factor at the ends of the branches is prime.
Suggest
- One useful factor pair is (9 \times 10), since each factor can be broken down into primes: (9 = 3 \times 3) and (10 = 2 \times 5).
- Multiply the factors: (2^2 \times 3 \times 5 = 2 \times 2 \times 3 \times 5 = 60). Check that every factor is prime.
Imagine
- One possible tree starts with (60 = 6 \times 10), then splits (6 = 2 \times 3) and (10 = 2 \times 5). The prime factors are (2, 2, 3,) and (5).
- One strategy is to test prime numbers as divisors, starting with 2, then 3, 5, and so on. Keep dividing by prime numbers until the quotient is 1.
Share
- A factor is a whole number that divides another number evenly. A prime factor is a factor that is also prime.
- One possible mistake is stopping when a factor is composite. Help correct it by splitting that factor into smaller factors until all the factors are prime.