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# help pls

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Determine the smallest positive integer n such that 5^n equivalent n^5 mod 3.

Apr 14, 2024

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To solve this congruence, we need to find the smallest positive integer $$n$$ such that $$5^n \equiv n^5 \pmod{3}$$.

First, let's observe that $$5 \equiv 2 \pmod{3}$$. Therefore, $$5^n \equiv 2^n \pmod{3}$$.

Now, let's calculate the values of $$2^n$$ modulo 3 for small values of $$n$$:

- $$2^1 \equiv 2 \pmod{3}$$

- $$2^2 \equiv 1 \pmod{3}$$

- $$2^3 \equiv 2 \pmod{3}$$

- $$2^4 \equiv 1 \pmod{3}$$

From this, we can see a pattern emerge: the value of $$2^n$$ alternates between 1 and 2 modulo 3, with period 2.

Now, let's consider $$n^5$$. Since we're taking $$n^5$$ modulo 3, we can reduce $$n^5$$ to its residue modulo 3:

- $$1^5 \equiv 1 \pmod{3}$$

- $$2^5 \equiv 32 \equiv 2 \pmod{3}$$

- $$3^5 \equiv 243 \equiv 0 \pmod{3}$$

The pattern for $$n^5$$ modulo 3 is not as obvious as $$2^n$$, but we can see that for $$n = 2$$, $$n^5$$ matches $$2^n$$ modulo 3.

So, the smallest positive integer $$n$$ such that $$5^n \equiv n^5 \pmod{3}$$ is $$n = 2$$.

Apr 14, 2024