I am to find all possible values of a and b where a, b are positive

integers and to show that I have found all of them.For example:
"2^3 = 3^1 + 5" is a satisfied equation where(a, b) are (3,1).
I know that it can be solved using modular arithmetic but I am NEW to the concept so I would honestly appreciate if you can explain me the same example with the most basic steps in modular arithmetic.Awaiting your reply.Thank you.

To find all possible values of a and b that satisfy the equation "2^3 = 3^1 + 5", we can approach it using modular arithmetic.

Step 1: Determine the congruence relations
We want to find positive integer solutions for the equation. In modular arithmetic, the congruence relation gives a remainder when dividing a number by another number. In this case, we'll use modulo 4 since the number 2 is involved.

2^3 = 3^1 + 5 (mod 4)

Step 2: Evaluate each term modulo 4
We'll evaluate each term separately and then combine them using modular arithmetic.

Starting with the left-hand side (LHS):
The number 2 raised to any positive integer power alternates between 2 and 0 when taken modulo 4.
2^3 (mod 4) = 2 (mod 4) = 2

Now, let's move on to the right-hand side (RHS). Keep in mind that we are evaluating each term modulo 4:

3^1 (mod 4) = 3 (mod 4) = 3
5 (mod 4) = 1 (mod 4) = 1

Step 3: Rewrite the equation with modular arithmetic

2 (mod 4) = 3 + 1 (mod 4)

Step 4: Simplify the equation further
Since both sides are in modulo 4, we can simplify the equation even more:

2 ≡ 3 + 1 (mod 4)

Step 5: Check all possible values of a and b
To find all possible values of a and b, we need to check all possible remainders when divided by 4.

a ≡ 3 (mod 4)
b ≡ 1 (mod 4)

Starting with a, we can list some positive integer values:
a = 3, 7, 11, 15, 19...

Next, check for the corresponding values of b:
For a = 3, b = 1 + 4n, where n is a non-negative integer. So, we have one solution: (3, 1).

Step 6: Conclusion
The only solution for positive integers a and b that satisfy the equation "2^3 = 3^1 + 5" is (a, b) = (3, 1).

By following these steps, you can apply modular arithmetic to solve similar equations and find all possible values of a and b satisfying given conditions.