Find the remainder when 413 divided by 3?

To find the remainder when a large power like 4^13 is divided by 3, you can use modular arithmetic to simplify the base.

Step-by-Step Calculation

1. Simplify the base:

• The base is 4. When 4 is divided by 3, the remainder is 1.

• In modular notation, we write this as: 4 is congruent to 1 mod 3.

2. Apply the exponent:

• Since 4 is congruent to 1 mod 3, then any power of 4 will also be congruent to that same power of 1.

• 4^13 is congruent to 1^13 mod 3.

3. Calculate the final result:

• Since 1^13 = 1, the expression simplifies to 4^13 is congruent to 1 mod 3.

• This means that when 4^13 is divided by 3, the remainder is 1.