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1. Start by dividing the number by the smallest prime number, which is 2.

2. Determine how many times 2 divides into 256 completely.

- 256 ÷ 2 = 128

- 128 ÷ 2 = 64

- 64 ÷ 2 = 32

- 32 ÷ 2 = 16

- 16 ÷ 2 = 8

- 8 ÷ 2 = 4

- 4 ÷ 2 = 2

- 2 ÷ 2 = 1

3. After continuously dividing by 2, we finally reach 1, which means no further division is possible by 2.

By examining the results of each division, we see that 256 can be divided by 2 exactly 8 times. Therefore, we express 256 as:

[tex]\[ 256 = 2^8 \][/tex]

Thus, the prime factorization of 256 is:

[tex]\[ 256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \][/tex]

Or more compactly:

[tex]\[ 256 = 2^8 \][/tex]

So the prime factorization of 256 is [tex]\( 2^8 \)[/tex].