Discrete mathematics i 5. find all integers that satisfy the congruence relation 33x ≡ 38 (mod 280) .

ANSWER

The general solution is , where is an  integer

EXPLANATION

In order  to solve the linear congruence;

We need to determine the inverse of (which is a Bézout coefficient for 33).

To do that we must first use  the Euclidean Algorithm to verify the existence of the inverse by showing that;

Now, here we go;

The greatest common divisor is the last remainder before the remainder of zero.

Hence, the .

We now express this gcd of 1 as a linear combination of  33 and 280.

We can achieve this by making all the non zero remainders the subject and making a backward substitution.

Equation (2) in equation (1) gives,

The above linear combination tells us that   is the inverse of .

Now we multiply both sides of our congruence relation by .

This implies that;

.

Since this is modulo, the solution is not unique because any integral addition or subtraction of the modulo (280 in this case) produces an equivalent solution.

Therefore the general solution is,

, where is an  integer

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