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 |
|---|---|---|---|---|---|---|
| 293 | 444 | 0 | 293 | 0 | 1 | 0 |
| 444 | 293 | 1 | 151 | 1 | 0 | 1 |
| 293 | 151 | 1 | 142 | 0 | 1 | -1 |
| 151 | 142 | 1 | 9 | 1 | -1 | 2 |
| 142 | 9 | 15 | 7 | -1 | 2 | -31 |
| 9 | 7 | 1 | 2 | 2 | -31 | 33 |
| 7 | 2 | 3 | 1 | -31 | 33 | -130 |
| 2 | 1 | 2 | 0 | 33 | -130 | 293 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 398 | 397 | 1 | 1 | 0 | 1 | -1 |
| 397 | 1 | 397 | 0 | 1 | -1 | 398 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 813 | 253 | 3 | 54 | 0 | 1 | -3 |
| 253 | 54 | 4 | 37 | 1 | -3 | 13 |
| 54 | 37 | 1 | 17 | -3 | 13 | -16 |
| 37 | 17 | 2 | 3 | 13 | -16 | 45 |
| 17 | 3 | 5 | 2 | -16 | 45 | -241 |
| 3 | 2 | 1 | 1 | 45 | -241 | 286 |
| 2 | 1 | 2 | 0 | -241 | 286 | -813 |
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 ≡ 787 × 444-1 (mod 293) ≡ 787 × 163 (mod 293) ≡ 240 (mod 293)x ≡ 546 × 397-1 (mod 398) ≡ 546 × 397 (mod 398) ≡ 250 (mod 398)x ≡ 402 × 253-1 (mod 813) ≡ 402 × 286 (mod 813) ≡ 339 (mod 813)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 240 (mod 293)
x ≡ 250 (mod 398)
x ≡ 339 (mod 813)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 293 × 398 × 813 = 94807182 - We calculate the numbers M1 to M3
M1=M/m1=94807182/293=323574,M2=M/m2=94807182/398=238209,M3=M/m3=94807182/813=116614 - 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 -135 mod 293 ≡ 158n b q r t1 t2 t3 293 323574 0 293 0 1 0 323574 293 1104 102 1 0 1 293 102 2 89 0 1 -2 102 89 1 13 1 -2 3 89 13 6 11 -2 3 -20 13 11 1 2 3 -20 23 11 2 5 1 -20 23 -135 2 1 2 0 23 -135 293
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -33 mod 398 ≡ 365n b q r t1 t2 t3 398 238209 0 398 0 1 0 238209 398 598 205 1 0 1 398 205 1 193 0 1 -1 205 193 1 12 1 -1 2 193 12 16 1 -1 2 -33 12 1 12 0 2 -33 398
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -371 mod 813 ≡ 442n b q r t1 t2 t3 813 116614 0 813 0 1 0 116614 813 143 355 1 0 1 813 355 2 103 0 1 -2 355 103 3 46 1 -2 7 103 46 2 11 -2 7 -16 46 11 4 2 7 -16 71 11 2 5 1 -16 71 -371 2 1 2 0 71 -371 813
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
= (240 × 323574 × 158 +
250 × 238209 × 365 +
339 × 116614 × 442) mod 94807182
= 94213218 (mod 94807182)
So our answer is 94213218 (mod 94807182).
Verification
So we found that x ≡ 94213218
If this is correct, then the following statements (i.e. the original equations) are true:
444x (mod 293) ≡ 787 (mod 293)
397x (mod 398) ≡ 546 (mod 398)
253x (mod 813) ≡ 402 (mod 813)
Let's see whether that's indeed the case if we use x ≡ 94213218.