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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

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