Question

The greatest prime factor of \((3^{199} - 3^{196})\) is:

• A. 13
• B. 17
• C. 3
• D. 11

Hint:

When dealing with exponential expressions, factor out common terms and analyze the prime factorization of constants to find the greatest prime factor efficiently.

Answer 1

The Correct Option is A

Step 1: Factor the given expression.
The given expression can be rewritten by factoring out the common term: \[ 3^{199} - 3^{196} = 3^{196} \cdot (3^3 - 1) \] Step 2: Evaluate \(3^3 - 1\).
\[ 3^3 = 27 \quad {thus,} \quad 3^3 - 1 = 27 - 1 = 26 \] Substituting the value back, the expression simplifies to: \[ 3^{199} - 3^{196} = 3^{196} \cdot 26 \] Step 3: Prime factorization of 26.
Breaking 26 into its prime factors: \[ 26 = 2 \times 13 \] Step 4: Determine the highest prime factor.
The complete set of prime factors in the expression includes \(3, 2, {and } 13\). The largest prime factor is: \[ \boxed{13} \] Final Answer: \[ \boxed{{(A) 13}} \]