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

 Apr 14, 2024
 #1
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To solve this congruence, we need to find the smallest positive integer n such that 5nn5(mod3).

 

First, let's observe that 52(mod3). Therefore, 5n2n(mod3).

 

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

 

- 212(mod3)


- 221(mod3)


- 232(mod3)


- 241(mod3)

 

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

 

Now, let's consider n5. Since we're taking n5 modulo 3, we can reduce n5 to its residue modulo 3:

 

- 151(mod3)


- 25322(mod3)


- 352430(mod3)

 

The pattern for n5 modulo 3 is not as obvious as 2n, but we can see that for n=2, n5 matches 2n modulo 3.

 

So, the smallest positive integer n such that 5nn5(mod3) is n=2.

 Apr 14, 2024

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