Remainder when 17^23 is divided by 16 will be ____

Step-by-Step Solution

1. Analyze the Base: We want to find the remainder of 17^23 when divided by 16.

2. Use Modular Arithmetic: First, look at the base, 17, relative to the divisor, 16.

   • 17 divided by 16 leaves a remainder of 1.

   • In modular notation, this is written as: 17 ≡ 1 (mod 16).

3. Apply the Exponent: Since 17 is congruent to 1 modulo 16, then any power of 17 is also congruent to that same power of 1 modulo 16.

   • 17^23 ≡ 1^23 (mod 16).

4. Simplify: 1 raised to any power is 1.

   • 1^23 = 1.

5. Conclusion: The remainder when 17^23 is divided by 16 is 1.