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Let x and y be integers. If x and y satisfy 41x + 5y = 31, then find the residue of x modulo 5.
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You are given this equation
41x + 5y = 31, (1)
where x and y are integers. Hence
41x = 31 - 5y. (2)
They want you find residue {x mod 5} from modular equation (2).
It means that you consider equation (2) as an equivalence by modulo 5.
If so, you may forget about the term -5y, because it is 0 (zero) by the modulo 5.
So, your task is to find the residue {x mod 5} from the modular equation
41x = 31 mod 5,
which is equivalent to
41x = 1 mod 5. (3)
The set of possible solutions is x = {0 mod 5}, {1 mod 5}, {2 mod 5}, {3 mod 5} and {4 mod 5}.
x = {0 mod 5} turns (3) into 41*0 = 1 mod 5, which is obviously wrong.
x = {1 mod 5} turns (3) into 41*1 = 1 mod 5, which is obviously correct modular equation ( since 41 mod 5 = 1 mod 5).
x = {2 mod 5} turns (3) into 41*2 = 1 mod 5, which is obviously wrong.
x = {3 mod 5} turns (3) into 41*3 = 1 mod 5, which is obviously wrong.
x = {4 mod 5} turns (3) into 41*4 = 1 mod 5, which is obviously wrong.
Thus of five possible solutions only one is valid: x = {1 mod 5}. ANSWER
So, in this problem, you can easily find the answer by the brute force method.
The problem can be solved using more refined methods, but this reasoning is the number 1,
which you should know and understand as a basic method to start thinking.