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 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 |
|---|---|---|---|---|---|---|
| 323 | 805 | 0 | 323 | 0 | 1 | 0 |
| 805 | 323 | 2 | 159 | 1 | 0 | 1 |
| 323 | 159 | 2 | 5 | 0 | 1 | -2 |
| 159 | 5 | 31 | 4 | 1 | -2 | 63 |
| 5 | 4 | 1 | 1 | -2 | 63 | -65 |
| 4 | 1 | 4 | 0 | 63 | -65 | 323 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 491 | 408 | 1 | 83 | 0 | 1 | -1 |
| 408 | 83 | 4 | 76 | 1 | -1 | 5 |
| 83 | 76 | 1 | 7 | -1 | 5 | -6 |
| 76 | 7 | 10 | 6 | 5 | -6 | 65 |
| 7 | 6 | 1 | 1 | -6 | 65 | -71 |
| 6 | 1 | 6 | 0 | 65 | -71 | 491 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 533 | 789 | 0 | 533 | 0 | 1 | 0 |
| 789 | 533 | 1 | 256 | 1 | 0 | 1 |
| 533 | 256 | 2 | 21 | 0 | 1 | -2 |
| 256 | 21 | 12 | 4 | 1 | -2 | 25 |
| 21 | 4 | 5 | 1 | -2 | 25 | -127 |
| 4 | 1 | 4 | 0 | 25 | -127 | 533 |
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 ≡ 301 × 805-1 (mod 323) ≡ 301 × 258 (mod 323) ≡ 138 (mod 323)x ≡ 499 × 408-1 (mod 491) ≡ 499 × 420 (mod 491) ≡ 414 (mod 491)x ≡ 891 × 789-1 (mod 533) ≡ 891 × 406 (mod 533) ≡ 372 (mod 533)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 138 (mod 323)
x ≡ 414 (mod 491)
x ≡ 372 (mod 533)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 323 × 491 × 533 = 84530069 - We calculate the numbers M1 to M3
M1=M/m1=84530069/323=261703,M2=M/m2=84530069/491=172159,M3=M/m3=84530069/533=158593 - 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 -146 mod 323 ≡ 177n b q r t1 t2 t3 323 261703 0 323 0 1 0 261703 323 810 73 1 0 1 323 73 4 31 0 1 -4 73 31 2 11 1 -4 9 31 11 2 9 -4 9 -22 11 9 1 2 9 -22 31 9 2 4 1 -22 31 -146 2 1 2 0 31 -146 323
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 116 mod 491 ≡ 116n b q r t1 t2 t3 491 172159 0 491 0 1 0 172159 491 350 309 1 0 1 491 309 1 182 0 1 -1 309 182 1 127 1 -1 2 182 127 1 55 -1 2 -3 127 55 2 17 2 -3 8 55 17 3 4 -3 8 -27 17 4 4 1 8 -27 116 4 1 4 0 -27 116 -491
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 115 mod 533 ≡ 115n b q r t1 t2 t3 533 158593 0 533 0 1 0 158593 533 297 292 1 0 1 533 292 1 241 0 1 -1 292 241 1 51 1 -1 2 241 51 4 37 -1 2 -9 51 37 1 14 2 -9 11 37 14 2 9 -9 11 -31 14 9 1 5 11 -31 42 9 5 1 4 -31 42 -73 5 4 1 1 42 -73 115 4 1 4 0 -73 115 -533
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
= (138 × 261703 × 177 +
414 × 172159 × 116 +
372 × 158593 × 115) mod 84530069
= 58622377 (mod 84530069)
So our answer is 58622377 (mod 84530069).
Verification
So we found that x ≡ 58622377
If this is correct, then the following statements (i.e. the original equations) are true:
805x (mod 323) ≡ 301 (mod 323)
408x (mod 491) ≡ 499 (mod 491)
789x (mod 533) ≡ 891 (mod 533)
Let's see whether that's indeed the case if we use x ≡ 58622377.