what is the prime factorization of 64
The Correct Answer and Explanation is:
The prime factorization of 64 is:
2 × 2 × 2 × 2 × 2 × 2
Or using exponents: 2⁶
Explanation:
Prime factorization is the process of expressing a number as a product of its prime factors. A prime number has only two distinct positive divisors: 1 and itself. The number 2 is the smallest and only even prime number. Every even number can be divided by 2, which makes 2 an essential factor in prime factorization of even numbers.
The number 64 is even, so division by 2 begins the process. Dividing 64 by 2 gives 32. Repeating this division step:
64 ÷ 2 = 32
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
After seven total divisions by 2, the result is 1. Each step used the same prime factor, which is 2. Since all factors are 2 and no other prime number is involved, 64 is a perfect power of 2.
Multiplying the factors confirms this:
2 × 2 × 2 × 2 × 2 × 2 = 64
This product shows the number 2 is used six times. In exponential form, the factorization becomes 2 raised to the power of 6. This representation is often preferred because it is more compact and highlights the number of times the prime factor appears.
Prime factorization helps in simplifying fractions, finding least common multiples, greatest common divisors, and solving many mathematical problems involving integers. The uniqueness of prime factorization is guaranteed by the Fundamental Theorem of Arithmetic, which states that every positive integer greater than 1 has a unique prime factorization, regardless of the order of the factors.
In summary, 64 is expressed as a product of the same prime number, 2, repeated six times, resulting in the prime factorization 2⁶
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