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.
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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 |
|---|---|---|---|---|---|---|
| 966 | 523 | 1 | 443 | 0 | 1 | -1 |
| 523 | 443 | 1 | 80 | 1 | -1 | 2 |
| 443 | 80 | 5 | 43 | -1 | 2 | -11 |
| 80 | 43 | 1 | 37 | 2 | -11 | 13 |
| 43 | 37 | 1 | 6 | -11 | 13 | -24 |
| 37 | 6 | 6 | 1 | 13 | -24 | 157 |
| 6 | 1 | 6 | 0 | -24 | 157 | -966 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 491 | 573 | 0 | 491 | 0 | 1 | 0 |
| 573 | 491 | 1 | 82 | 1 | 0 | 1 |
| 491 | 82 | 5 | 81 | 0 | 1 | -5 |
| 82 | 81 | 1 | 1 | 1 | -5 | 6 |
| 81 | 1 | 81 | 0 | -5 | 6 | -491 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 415 | 457 | 0 | 415 | 0 | 1 | 0 |
| 457 | 415 | 1 | 42 | 1 | 0 | 1 |
| 415 | 42 | 9 | 37 | 0 | 1 | -9 |
| 42 | 37 | 1 | 5 | 1 | -9 | 10 |
| 37 | 5 | 7 | 2 | -9 | 10 | -79 |
| 5 | 2 | 2 | 1 | 10 | -79 | 168 |
| 2 | 1 | 2 | 0 | -79 | 168 | -415 |
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 ≡ 721 × 523-1 (mod 966) ≡ 721 × 157 (mod 966) ≡ 175 (mod 966)x ≡ 912 × 573-1 (mod 491) ≡ 912 × 6 (mod 491) ≡ 71 (mod 491)x ≡ 588 × 457-1 (mod 415) ≡ 588 × 168 (mod 415) ≡ 14 (mod 415)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 175 (mod 966)
x ≡ 71 (mod 491)
x ≡ 14 (mod 415)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 966 × 491 × 415 = 196836990 - We calculate the numbers M1 to M3
M1=M/m1=196836990/966=203765,M2=M/m2=196836990/491=400890,M3=M/m3=196836990/415=474306 - 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 95 mod 966 ≡ 95n b q r t1 t2 t3 966 203765 0 966 0 1 0 203765 966 210 905 1 0 1 966 905 1 61 0 1 -1 905 61 14 51 1 -1 15 61 51 1 10 -1 15 -16 51 10 5 1 15 -16 95 10 1 10 0 -16 95 -966
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 128 mod 491 ≡ 128n b q r t1 t2 t3 491 400890 0 491 0 1 0 400890 491 816 234 1 0 1 491 234 2 23 0 1 -2 234 23 10 4 1 -2 21 23 4 5 3 -2 21 -107 4 3 1 1 21 -107 128 3 1 3 0 -107 128 -491
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -149 mod 415 ≡ 266n b q r t1 t2 t3 415 474306 0 415 0 1 0 474306 415 1142 376 1 0 1 415 376 1 39 0 1 -1 376 39 9 25 1 -1 10 39 25 1 14 -1 10 -11 25 14 1 11 10 -11 21 14 11 1 3 -11 21 -32 11 3 3 2 21 -32 117 3 2 1 1 -32 117 -149 2 1 2 0 117 -149 415
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
= (175 × 203765 × 95 +
71 × 400890 × 128 +
14 × 474306 × 266) mod 196836990
= 136369429 (mod 196836990)
So our answer is 136369429 (mod 196836990).
Verification
So we found that x ≡ 136369429
If this is correct, then the following statements (i.e. the original equations) are true:
523x (mod 966) ≡ 721 (mod 966)
573x (mod 491) ≡ 912 (mod 491)
457x (mod 415) ≡ 588 (mod 415)
Let's see whether that's indeed the case if we use x ≡ 136369429.