Use Fermat's Little Theorem to �find the remainder on division of 5120 by 19.
To use Fermat's Little Theorem to find the remainder on division of a number by another number, follow these steps:
1. Start with the given number and the modulus:
- Number: 5120
- Modulus: 19
2. Check if the modulus is a prime number. In this case, 19 is a prime number, so Fermat's Little Theorem can be applied.
3. Apply Fermat's Little Theorem:
- The theorem states that if p is a prime number, and a is any integer not divisible by p, then a^(p-1) ≡ 1 (mod p). Here, "^" denotes exponentiation and "≡" denotes congruence.
4. Now, substitute the values into the theorem:
- p = 19
- a = 5120
- 5120^(19-1) ≡ 1 (mod 19)
5. Simplify the exponent:
- 5120^18 ≡ 1 (mod 19)
6. Calculate the remainder:
- Raise 5120 to the power of 18 and divide the result by 19. The remainder you get is the answer to the problem.
In this case, using a calculator or programming tool, you can find the result to be 1.
Therefore, the remainder on division of 5120 by 19 is 1.