Prime Factorization - Pre Algebra

1. Using prime factorizations, what is the LCM of $2^3 \cdot 3$ and $2^2 \cdot 3^2$?

   $72$. Take the larger exponents: $2^3 \cdot 3^2$.

2. What is the prime factorization of a whole number?

   The number written as a product of prime numbers only, for example $12 = 2^2 \cdot 3$.

3. What number has the prime factorization $2 \cdot 3^2 \cdot 5$?

   $90$. Compute $2 \cdot 9 \cdot 5$.

4. What is the prime factorization of $84$?

   $2^2 \cdot 3 \cdot 7$. Since $84 = 4 \cdot 21$.

5. What is the prime factorization of $36$?

   $2^2 \cdot 3^2$. Since $36 = 4 \cdot 9$.

6. What is the prime factorization of $60$?

   $2^2 \cdot 3 \cdot 5$. Since $60 = 4 \cdot 15$.

7. What is the prime factorization of $45$?

   $3^2 \cdot 5$. Since $45 = 9 \cdot 5$.

8. What is the prime factorization of $100$?

   $2^2 \cdot 5^2$. Since $100 = 4 \cdot 25$.

9. How do you find the GCF of two numbers from their prime factorizations?

   Multiply the primes they share, each raised to the smaller of its two exponents.

10. How many different prime factorizations can a whole number greater than $1$ have, ignoring order?

    Exactly one. This is the Fundamental Theorem of Arithmetic.

11. Using prime factorizations, what is the GCF of $2^3 \cdot 3$ and $2^2 \cdot 3^2$?

    $12$. Take the smaller exponents: $2^2 \cdot 3$.

12. How do you find the LCM of two numbers from their prime factorizations?

    Multiply every prime that appears in either number, each raised to the larger of its exponents.

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