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 |
|---|---|---|---|---|---|---|
| 557 | 71 | 7 | 60 | 0 | 1 | -7 |
| 71 | 60 | 1 | 11 | 1 | -7 | 8 |
| 60 | 11 | 5 | 5 | -7 | 8 | -47 |
| 11 | 5 | 2 | 1 | 8 | -47 | 102 |
| 5 | 1 | 5 | 0 | -47 | 102 | -557 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 971 | 832 | 1 | 139 | 0 | 1 | -1 |
| 832 | 139 | 5 | 137 | 1 | -1 | 6 |
| 139 | 137 | 1 | 2 | -1 | 6 | -7 |
| 137 | 2 | 68 | 1 | 6 | -7 | 482 |
| 2 | 1 | 2 | 0 | -7 | 482 | -971 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 547 | 327 | 1 | 220 | 0 | 1 | -1 |
| 327 | 220 | 1 | 107 | 1 | -1 | 2 |
| 220 | 107 | 2 | 6 | -1 | 2 | -5 |
| 107 | 6 | 17 | 5 | 2 | -5 | 87 |
| 6 | 5 | 1 | 1 | -5 | 87 | -92 |
| 5 | 1 | 5 | 0 | 87 | -92 | 547 |
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 ≡ 501 × 71-1 (mod 557) ≡ 501 × 102 (mod 557) ≡ 415 (mod 557)x ≡ 629 × 832-1 (mod 971) ≡ 629 × 482 (mod 971) ≡ 226 (mod 971)x ≡ 170 × 327-1 (mod 547) ≡ 170 × 455 (mod 547) ≡ 223 (mod 547)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 415 (mod 557)
x ≡ 226 (mod 971)
x ≡ 223 (mod 547)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 557 × 971 × 547 = 295843309 - We calculate the numbers M1 to M3
M1=M/m1=295843309/557=531137,M2=M/m2=295843309/971=304679,M3=M/m3=295843309/547=540847 - 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 104 mod 557 ≡ 104n b q r t1 t2 t3 557 531137 0 557 0 1 0 531137 557 953 316 1 0 1 557 316 1 241 0 1 -1 316 241 1 75 1 -1 2 241 75 3 16 -1 2 -7 75 16 4 11 2 -7 30 16 11 1 5 -7 30 -37 11 5 2 1 30 -37 104 5 1 5 0 -37 104 -557
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 140 mod 971 ≡ 140n b q r t1 t2 t3 971 304679 0 971 0 1 0 304679 971 313 756 1 0 1 971 756 1 215 0 1 -1 756 215 3 111 1 -1 4 215 111 1 104 -1 4 -5 111 104 1 7 4 -5 9 104 7 14 6 -5 9 -131 7 6 1 1 9 -131 140 6 1 6 0 -131 140 -971
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -181 mod 547 ≡ 366n b q r t1 t2 t3 547 540847 0 547 0 1 0 540847 547 988 411 1 0 1 547 411 1 136 0 1 -1 411 136 3 3 1 -1 4 136 3 45 1 -1 4 -181 3 1 3 0 4 -181 547
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
= (415 × 531137 × 104 +
226 × 304679 × 140 +
223 × 540847 × 366) mod 295843309
= 83349895 (mod 295843309)
So our answer is 83349895 (mod 295843309).
Verification
So we found that x ≡ 83349895
If this is correct, then the following statements (i.e. the original equations) are true:
71x (mod 557) ≡ 501 (mod 557)
832x (mod 971) ≡ 629 (mod 971)
327x (mod 547) ≡ 170 (mod 547)
Let's see whether that's indeed the case if we use x ≡ 83349895.