Determine the smallest positive integer n such that 5^n equivalent n^5 mod 3.
To solve this congruence, we need to find the smallest positive integer n such that 5n≡n5(mod3).
First, let's observe that 5≡2(mod3). Therefore, 5n≡2n(mod3).
Now, let's calculate the values of 2n modulo 3 for small values of n:
- 21≡2(mod3)
- 22≡1(mod3)
- 23≡2(mod3)
- 24≡1(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:
- 15≡1(mod3)
- 25≡32≡2(mod3)
- 35≡243≡0(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 5n≡n5(mod3) is n=2.