Determine whether or not the following congruence has a natural number solution

Question

Determine whether or not the following congruence has a natural number solution: $$5^x + 3 \equiv 5 \mod 100$$

Answer 1

Since x is $5^x+3≅5\quad mod\ \ 100$ therefore $5^x+3≅5\quad mod\ \ 5$. This means that$$\exists k\in\Bbb Z\qquad ,\qquad 5^x+3=5k\to5(k-5^{x-1})=3$$since $x\in\Bbb N$ therefore $k-5^{x-1}\in\Bbb Z$ which leads to $5|3$ and this is a contradiction. So this equation has to answer in natural numbers.

Answer 2

$5^{x+3}$ is ended on $25$, which says that our statement is wrong.

If you mean $5^x+3$ then it's ended on $28$ and it's still impossible.

Answer 3

This congruence implies $ 5^x \equiv 2\mod 100$. Now $5$ to the power anything has $5$ in the units digit. The least positive representative modulo $100$ of such an integer is $5$. So the equation does not admit a natural number solution.