Write the prime factorization of each number. 64
The Correct Answer and Explanation is:
Prime Factorization of 64:
64 = 2 × 2 × 2 × 2 × 2 × 2 = 2⁶
Explanation:
Prime factorization involves expressing a number as a product of its prime factors. A prime number has only two positive divisors: 1 and itself. The number 2 is the smallest and only even prime number. In this case, 64 is a composite number, meaning it can be broken down into smaller prime numbers multiplied together.
To find the prime factorization of 64, divide the number by the smallest prime number repeatedly until only prime numbers remain. Start with 64 and divide by 2, the smallest prime:
- 64 ÷ 2 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1
Each division uses the prime number 2. This process results in six 2s multiplied together, confirming that 64 is equal to 2 raised to the power of 6, or 2⁶. No other prime numbers are involved in the factorization of 64.
This form of factorization helps in many mathematical applications such as finding the greatest common divisor (GCD), least common multiple (LCM), simplifying fractions, and solving exponential equations. Understanding how to decompose numbers into their prime factors supports deeper understanding of number properties and divisibility rules.
The prime factorization process always results in a unique combination of prime numbers for any given positive integer greater than 1, which is known as the Fundamental Theorem of Arithmetic. This ensures that each number has only one prime factorization, regardless of the order in which the factors are written.
