Problem 26 - Introductory Algebra for College Students

Question

Identify each natural number as prime or composite. If the number is composite, find its prime factorization. $$64$$

Step-by-Step Solution

Answer

64 is a composite number and its prime factorization is \(2^6\).

Step 1: Identifying Prime or Composite

First identify if the number 64 is prime or composite. Since 64 has more than two distinct divisors (1, 64, 2, 32, 4, 16, 8), it is a composite number

Step 2: Finding Prime Factorization

Next, find its prime factorization. Start by dividing the number by the smallest prime number, which is 2. The result is \( 64 ÷ 2 = 32 \). Divide the result again by 2 to get \( 32 ÷ 2 = 16 \). Continue this process until you reach a number that isn't divisible by 2 anymore. Doing this, you get \(16 ÷ 2 = 8\), \(8 ÷ 2 = 4\), \(4 ÷ 2 = 2\), \(2 ÷ 2 = 1\). So, the prime factorization of 64 is \(2^6\) meaning \(64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2\)

Step 3: Conclusion

Therefore, 64 is a composite number and its prime factorization is \(2^6\)