Factorization of Expression

We need to find the greatest prime factor of the expression $(3^{199} – 3^{196})$. First, factor out the common term, which is the lowest power of 3, $3^{196}$.

$3^{199} – 3^{196} = 3^{196} (3^{199-196} – 1)$

Simplifying the Expression

Now, simplify the term inside the parenthesis:

$3^{196} (3^3 – 1)$

Calculate $3^3$:

$3^3 = 3 \times 3 \times 3 = 27$

Substitute this back into the expression:

$3^{196} (27 – 1) = 3^{196} \times 26$

Prime Factorization

To find the greatest prime factor, we determine the prime factors of the simplified expression $3^{196} \times 26$.

• The prime factors of $3^{196}$ consist only of the prime number 3.
• Find the prime factors of 26: $26 = 2 \times 13$. Both 2 and 13 are prime numbers.

The prime factors of the entire expression $(3^{199} – 3^{196})$ are therefore 3, 2, and 13.

Greatest Prime Factor Identification

Comparing the prime factors {2, 3, 13}, the largest number is 13.

Thus, the greatest prime factor of $(3^{199} – 3^{196})$ is 13.