Respuesta :
Your teacher must have made a typo somewhere. The expression 5^(2n-1) is not divisible by 8.
The expression 5^(2n-1) is composed of 2n-1 copies of "5" multiplied together. The key here is that 5 is the only unique prime factor of this number.
On the other hand, 8 = 2*2*2 = 2^3 showing that 2 is the only prime factor for this number. At this point, we can see that there's no way that 5^(2n-1) is divisible by 8. It all comes down to the clash of the primes 5 and 2 not matching up, and having nothing in common.
Let's consider a few concrete examples:
- If n = 1, then 5^(2n-1) = 5^(2*1-1) = 5^1 = 5 which is not divisible by 8. The number 5/8 = 0.625 isn't an integer.
- If n = 2, then 5^(2n-1) = 5^(2*2-1) = 5^3 = 125 which is also not a multiple of 8. The number 125/8 = 15.625 isn't an integer.
I'll let you try other natural numbers for n, and you'll find that [tex]\frac{5^{2n-1}}{8}[/tex] isn't an integer.
