# How do I solve a modular equation (find all soloutions)?

I have a lot of equations of this form: (Please note that any variable is either 0 or 1, not any real number).

$2(b+d) + 1(a+c) = 0 (mod 3)$

How do I go about finding all soloutions? I know this is largely related to number theory but where can I learn it? Here are a few instant ones )b=1, d=1, a=1, c=1, a=0), and (a=0, b=0, c=0,d=0), but how can I efficiently find all of them?

Also what do I do if

$2(b+d) + 1(a+c) + 2 = 0 ( mod 3)$

?

## 1 Answer

To find all solutions for the equation 2(b+d) + 1(a+c) ≡ 0 (mod 3) where the variables can only take the values 0 or 1, you can approach it using modular arithmetic.

Since the equation is congruent to 0 modulo 3, you can analyze the possible values of each term modulo 3. Let's consider each term separately:

For 2(b+d), the possible values modulo 3 are 0, 2 (since 2 ≡ 2 (mod 3)).

For 1(a+c), the possible values modulo 3 are 0, 1 (since 1 ≡ 1 (mod 3)).

Now, we can consider all possible combinations of these values for the terms and see which ones satisfy the equation.

The four possible combinations for 2(b+d) are:

1. 2 ≡ 2 (mod 3)

2. 0 ≡ 0 (mod 3)

3. 2 ≡ 2 (mod 3)

4. 0 ≡ 0 (mod 3)

The two possible combinations for 1(a+c) are:

1. 0 ≡ 0 (mod 3)

2. 1 ≡ 1 (mod 3)

To satisfy the equation, the sum of the terms from each combination should be congruent to 0 modulo 3.

Taking the combinations into account, we have:

1. 2 + 0 ≡ 2 (mod 3) and 0 ≡ 0 (mod 3) → Solution: (b = 1, d = 0, a = 0, c = 0)

2. 0 + 0 ≡ 0 (mod 3) and 0 ≡ 0 (mod 3) → Solution: (b = 0, d = 0, a = 0, c = 0)

3. 2 + 2 ≡ 1 (mod 3) and 0 ≡ 0 (mod 3) → No solution

4. 0 + 0 ≡ 0 (mod 3) and 1 ≡ 1 (mod 3) → No solution

Therefore, the solutions for the given equation are:

1. (b = 1, d = 0, a = 0, c = 0)

2. (b = 0, d = 0, a = 0, c = 0)

If you have an additional term in the equation, such as 2(b+d) + 1(a+c) + 2 ≡ 0 (mod 3), you can follow a similar approach. Consider the possible values of the additional term modulo 3 (which are 2 and 1) and combine them with the possible values of the other terms as shown above.

To learn more about number theory and modular arithmetic, you can refer to textbooks such as "Elementary Number Theory" by David M. Burton or "Introduction to the Theory of Numbers" by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery. Online resources and video lectures on number theory are also available, including courses on platforms like Coursera or Khan Academy.

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This is a high level question, and deserves offering a bounty!

If the only values a,b,c,d can take are 0 or 1, there are at most 16 possible arrangements for theirv values. You could list them all and see in which cases the equation is satisfied.