Question

do you understand?

1. essential question how can you write the prime factorization and find the greatest common factor and the least common multiple of two numbers?
2. what are two different ways in which you can use prime factorization to find the prime factors of a number?
3. generalize why is the gcf of two prime numbers always 1?
4. construct arguments in example 4, grant finds applesauce that comes in packages of 8, but now he finds juice bottles in only packages of 3. will the lcm change? explain.
5. critique reasoning sarah says that you can find the lcm of any two whole numbers by multiplying them together. provide a counterexample to show that sarah’s statement is incorrect.

do you know how?
in 6–8, write the prime factorization of each number. if the number is prime, write prime.

1. 33
2. 32
3. 19

in 9–11, find the gcf for each pair of numbers.

1. 18, 36
2. 22, 55
3. 100, 48

in 12–14, find the lcm for each pair of numbers.

1. 2, 5
2. 8, 12
3. omitted as not fully visible

Explanation:

Let's solve question 6: Prime factorization of 33.

Step 1: Check divisibility by smallest primes

Start with the smallest prime number, 2. 33 is odd, so not divisible by 2. Next prime is 3. \( 33 \div 3 = 11 \).

Step 2: Check if quotient is prime

11 is a prime number (only divisible by 1 and itself). So the prime factors of 33 are 3 and 11.

Step 1: Divide by 2 repeatedly

32 is even, so divide by 2: \( 32 \div 2 = 16 \).

Step 2: Continue dividing by 2

\( 16 \div 2 = 8 \).

Step 3: Continue dividing by 2

\( 8 \div 2 = 4 \).

Step 4: Continue dividing by 2

\( 4 \div 2 = 2 \).

Step 5: Continue dividing by 2

\( 2 \div 2 = 1 \).
We divided by 2 five times. So the prime factorization of 32 is \( 2^5 \).

Step 1: Check divisibility by primes less than its square root

The square root of 19 is approximately 4.36. So we check primes less than 4.36: 2, 3.
19 is odd, so not divisible by 2. \( 19 \div 3 \approx 6.33 \), not an integer. So 19 has no divisors other than 1 and itself.

Answer:

The prime factorization of 33 is \( 3 \times 11 \).

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Let's solve question 7: Prime factorization of 32.