Question

Compute the smallest natural number number congruent to 13²⁰⁰(mod 6) .

Answer 1

The smallest natural number congruent to 13²⁰⁰(mod 6) is 1.

Step-by-step explanation:

To compute the smallest natural number congruent to 13²⁰⁰(mod 6), we need to find the remainder when 13²⁰⁰ is divided by 6.

We can simplify the calculation by looking for a pattern. Let's start by finding the remainders of 13 when divided by 6:

13 ÷ 6 = 2 remainder 1

So, the remainders of successive powers of 13 also follow a pattern:

13² ≡ (2 remainder 1) × (2 remainder 1) ≡ 4 remainder 1

13³ ≡ (4 remainder 1) × (2 remainder 1) ≡ 8 remainder 1

Continuing this pattern, we find that 13⁴ ≡ 2 remainder 1, and so on. Since the remainder is always 1, we can conclude that 13²⁰⁰ ≡ 1 (mod 6). Therefore, the smallest natural number congruent to 13²⁰⁰(mod 6) is 1.