Finding integral solution of an equation

Question

This is part of a bigger question. Its actually a mathematical problem. So it would be really great if someone can direct me to any algorithm to obtain the solution of this problem or a pseudo code will be of help.

The question. Given an equation check if it has an integral solution. For example:

(26a+5)/32=b

Here a is an integer. Is there an algorithm to predict or find if b can be an integer. I need a general solution not specific to this question. The equation can vary. Thanks

Answer 1

Your problem is an example of a linear Diophantine equation. About that, Wikipedia says:

This Diophantine equation [i.e., a x + b y = c] has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if (x, y) is a solution, then the other solutions have the form (x + k v, y - k u), where k is an arbitrary integer, and u and v are the quotients of a and b (respectively) by the greatest common divisor of a and b.

In this case, (26 a + 5)/32 = b is equivalent to 26 a - 32 b = -5. The gcd of the coefficients of the unknowns is gcd(26, -32) = 2. Since -5 is not a multiple of 2, there is no solution.

A general Diophantine equation is a polynomial in the unknowns, and can only be solved (if at all) by more complex methods. A web search might turn up specialized software for that problem.

Answer 2

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b this means a=z'c and b=z''c then this is Bézout's identity of the form

with a=z' and b=z'' and the equation has an infinite number of solutions. So instead of trial searching method you can check if c is the greatest common divisor (GCD) of a and b

If indeed a and b are multiples of c then x and y can be computed using extended Euclidean algorithm which finds integers x and y (one of which is typically negative) that satisfy Bézout's identity

(as a side note: this holds also for any other Euclidean domain, i.e. polynomial ring & every Euclidean domain is unique factorization domain). You can use Iterative Method to find these solutions:

Integral solution to equation `a + bx = c + dy`