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Number of Solutions for Math Equations: 3x + y = 5702

ACMer

You are asked to find the number of order pairs (x, y) such that, and x, y are natural numbers which also satisfy .

Mathematics

This shouldn’t be too complex. If we let where .

If we mod 3, we then have this:


.

So, all the solutions are where where .

Since so we have ,

.

We have 51 solutions because k can be from 0 to 50 inclusive.

Bruteforce

If you don’t like maths, then write a small piece of C++ code that verifies this via bruteforce all possibilities within the conditions:

int sum = 0;
for (int x = 1; x <= 2002; x ++) {
    for (int y = 1; y <= 2003 - x; y ++) {
        if (3 * x + y == 5702) sum ++;
    }
}
cout << sum << endl;

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