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" .
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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 |
|---|---|---|---|---|---|---|
| 575 | 777 | 0 | 575 | 0 | 1 | 0 |
| 777 | 575 | 1 | 202 | 1 | 0 | 1 |
| 575 | 202 | 2 | 171 | 0 | 1 | -2 |
| 202 | 171 | 1 | 31 | 1 | -2 | 3 |
| 171 | 31 | 5 | 16 | -2 | 3 | -17 |
| 31 | 16 | 1 | 15 | 3 | -17 | 20 |
| 16 | 15 | 1 | 1 | -17 | 20 | -37 |
| 15 | 1 | 15 | 0 | 20 | -37 | 575 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 331 | 640 | 0 | 331 | 0 | 1 | 0 |
| 640 | 331 | 1 | 309 | 1 | 0 | 1 |
| 331 | 309 | 1 | 22 | 0 | 1 | -1 |
| 309 | 22 | 14 | 1 | 1 | -1 | 15 |
| 22 | 1 | 22 | 0 | -1 | 15 | -331 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 349 | 872 | 0 | 349 | 0 | 1 | 0 |
| 872 | 349 | 2 | 174 | 1 | 0 | 1 |
| 349 | 174 | 2 | 1 | 0 | 1 | -2 |
| 174 | 1 | 174 | 0 | 1 | -2 | 349 |
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 ≡ 547 × 777-1 (mod 575) ≡ 547 × 538 (mod 575) ≡ 461 (mod 575)x ≡ 949 × 640-1 (mod 331) ≡ 949 × 15 (mod 331) ≡ 2 (mod 331)x ≡ 493 × 872-1 (mod 349) ≡ 493 × 347 (mod 349) ≡ 61 (mod 349)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 461 (mod 575)
x ≡ 2 (mod 331)
x ≡ 61 (mod 349)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 575 × 331 × 349 = 66423425 - We calculate the numbers M1 to M3
M1=M/m1=66423425/575=115519,M2=M/m2=66423425/331=200675,M3=M/m3=66423425/349=190325 - 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 154 mod 575 ≡ 154n b q r t1 t2 t3 575 115519 0 575 0 1 0 115519 575 200 519 1 0 1 575 519 1 56 0 1 -1 519 56 9 15 1 -1 10 56 15 3 11 -1 10 -31 15 11 1 4 10 -31 41 11 4 2 3 -31 41 -113 4 3 1 1 41 -113 154 3 1 3 0 -113 154 -575
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -119 mod 331 ≡ 212n b q r t1 t2 t3 331 200675 0 331 0 1 0 200675 331 606 89 1 0 1 331 89 3 64 0 1 -3 89 64 1 25 1 -3 4 64 25 2 14 -3 4 -11 25 14 1 11 4 -11 15 14 11 1 3 -11 15 -26 11 3 3 2 15 -26 93 3 2 1 1 -26 93 -119 2 1 2 0 93 -119 331
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 32 mod 349 ≡ 32n b q r t1 t2 t3 349 190325 0 349 0 1 0 190325 349 545 120 1 0 1 349 120 2 109 0 1 -2 120 109 1 11 1 -2 3 109 11 9 10 -2 3 -29 11 10 1 1 3 -29 32 10 1 10 0 -29 32 -349
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
= (461 × 115519 × 154 +
2 × 200675 × 212 +
61 × 190325 × 32) mod 66423425
= 22711236 (mod 66423425)
So our answer is 22711236 (mod 66423425).
Verification
So we found that x ≡ 22711236
If this is correct, then the following statements (i.e. the original equations) are true:
777x (mod 575) ≡ 547 (mod 575)
640x (mod 331) ≡ 949 (mod 331)
872x (mod 349) ≡ 493 (mod 349)
Let's see whether that's indeed the case if we use x ≡ 22711236.