Welcome to ChineseRemainderTheorem.com!
Use the calculator below to get a step-by-step calculation of the Chinese Remainder Theory. Just enter the numbers you would like and press "Calculate" .
This removes all numbers from the textboxes, such that you can fill in your own.
This fills all textboxes with random numbers. If you fill in random numbers yourself, it is very likely that those numbers do not have a solution. To avoid disappointment, use this button instead! It only uses random numbers that do have a solution.
This button is similar to the "Clear everything" button, but only clears the left column.
This is useful if you want your equations to be of the form x ≡ a (mod m) rather than bx ≡ a (mod m).
In that case, it can be especially useful after using the random numbers button.
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Are you ready to view a full step-by-step Chinese Remainder Theorem calculation for the numbers you have entered? Then use this button!
Want to know more?- Read about how to execute the Chinese Remainder Theorem
- Discover different ways to calculate multiplicative inverses
Transform the equations
You used one or more of the fields on the left, so your equations are of the form bx ≡ a mod m.
We want them to be of the form x ≡ a mod m, so we need to move the values on the left to the right side of the equation.
For a more detailed explanation about how this works, see this part of our page about how to execute the Chinese Remainder algorithm.
First, we calculate the inverses of the leftmost value on each row:
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 907 | 231 | 3 | 214 | 0 | 1 | -3 |
| 231 | 214 | 1 | 17 | 1 | -3 | 4 |
| 214 | 17 | 12 | 10 | -3 | 4 | -51 |
| 17 | 10 | 1 | 7 | 4 | -51 | 55 |
| 10 | 7 | 1 | 3 | -51 | 55 | -106 |
| 7 | 3 | 2 | 1 | 55 | -106 | 267 |
| 3 | 1 | 3 | 0 | -106 | 267 | -907 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 391 | 175 | 2 | 41 | 0 | 1 | -2 |
| 175 | 41 | 4 | 11 | 1 | -2 | 9 |
| 41 | 11 | 3 | 8 | -2 | 9 | -29 |
| 11 | 8 | 1 | 3 | 9 | -29 | 38 |
| 8 | 3 | 2 | 2 | -29 | 38 | -105 |
| 3 | 2 | 1 | 1 | 38 | -105 | 143 |
| 2 | 1 | 2 | 0 | -105 | 143 | -391 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 865 | 763 | 1 | 102 | 0 | 1 | -1 |
| 763 | 102 | 7 | 49 | 1 | -1 | 8 |
| 102 | 49 | 2 | 4 | -1 | 8 | -17 |
| 49 | 4 | 12 | 1 | 8 | -17 | 212 |
| 4 | 1 | 4 | 0 | -17 | 212 | -865 |
Source: ExtendedEuclideanAlgorithm.com
Click on any row to reveal a more detailed calculation of each multiplicative inverse.
Now that we now the inverses, let's move the leftmost value on each row to the right of the equation:
x ≡ 894 × 231-1 (mod 907) ≡ 894 × 267 (mod 907) ≡ 157 (mod 907)x ≡ 246 × 175-1 (mod 391) ≡ 246 × 143 (mod 391) ≡ 379 (mod 391)x ≡ 772 × 763-1 (mod 865) ≡ 772 × 212 (mod 865) ≡ 179 (mod 865)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 157 (mod 907)
x ≡ 379 (mod 391)
x ≡ 179 (mod 865)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 907 × 391 × 865 = 306761005 - We calculate the numbers M1 to M3
M1=M/m1=306761005/907=338215,M2=M/m2=306761005/391=784555,M3=M/m3=306761005/865=354637 - We now calculate the modular multiplicative inverses M1-1 to M3-1
Have a look at the page that explains how to calculate modular multiplicative inverse.
Using, for example, the Extended Euclidean Algorithm, we will find that:
So our multiplicative inverse is -274 mod 907 ≡ 633n b q r t1 t2 t3 907 338215 0 907 0 1 0 338215 907 372 811 1 0 1 907 811 1 96 0 1 -1 811 96 8 43 1 -1 9 96 43 2 10 -1 9 -19 43 10 4 3 9 -19 85 10 3 3 1 -19 85 -274 3 1 3 0 85 -274 907
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 58 mod 391 ≡ 58n b q r t1 t2 t3 391 784555 0 391 0 1 0 784555 391 2006 209 1 0 1 391 209 1 182 0 1 -1 209 182 1 27 1 -1 2 182 27 6 20 -1 2 -13 27 20 1 7 2 -13 15 20 7 2 6 -13 15 -43 7 6 1 1 15 -43 58 6 1 6 0 -43 58 -391
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 133 mod 865 ≡ 133n b q r t1 t2 t3 865 354637 0 865 0 1 0 354637 865 409 852 1 0 1 865 852 1 13 0 1 -1 852 13 65 7 1 -1 66 13 7 1 6 -1 66 -67 7 6 1 1 66 -67 133 6 1 6 0 -67 133 -865
Source: ExtendedEuclideanAlgorithm.com
- Now we can calculate x with the equation we saw earlier
x = (a1 × M1 × M1-1 + a2 × M2 × M2-1 + ... + ak × Mk × Mk-1) mod M
= (157 × 338215 × 633 +
379 × 784555 × 58 +
179 × 354637 × 133) mod 306761005
= 96202019 (mod 306761005)
So our answer is 96202019 (mod 306761005).
Verification
So we found that x ≡ 96202019
If this is correct, then the following statements (i.e. the original equations) are true:
231x (mod 907) ≡ 894 (mod 907)
175x (mod 391) ≡ 246 (mod 391)
763x (mod 865) ≡ 772 (mod 865)
Let's see whether that's indeed the case if we use x ≡ 96202019.