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 |
|---|---|---|---|---|---|---|
| 739 | 180 | 4 | 19 | 0 | 1 | -4 |
| 180 | 19 | 9 | 9 | 1 | -4 | 37 |
| 19 | 9 | 2 | 1 | -4 | 37 | -78 |
| 9 | 1 | 9 | 0 | 37 | -78 | 739 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 889 | 824 | 1 | 65 | 0 | 1 | -1 |
| 824 | 65 | 12 | 44 | 1 | -1 | 13 |
| 65 | 44 | 1 | 21 | -1 | 13 | -14 |
| 44 | 21 | 2 | 2 | 13 | -14 | 41 |
| 21 | 2 | 10 | 1 | -14 | 41 | -424 |
| 2 | 1 | 2 | 0 | 41 | -424 | 889 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 289 | 230 | 1 | 59 | 0 | 1 | -1 |
| 230 | 59 | 3 | 53 | 1 | -1 | 4 |
| 59 | 53 | 1 | 6 | -1 | 4 | -5 |
| 53 | 6 | 8 | 5 | 4 | -5 | 44 |
| 6 | 5 | 1 | 1 | -5 | 44 | -49 |
| 5 | 1 | 5 | 0 | 44 | -49 | 289 |
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 ≡ 324 × 180-1 (mod 739) ≡ 324 × 661 (mod 739) ≡ 593 (mod 739)x ≡ 909 × 824-1 (mod 889) ≡ 909 × 465 (mod 889) ≡ 410 (mod 889)x ≡ 225 × 230-1 (mod 289) ≡ 225 × 240 (mod 289) ≡ 246 (mod 289)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 593 (mod 739)
x ≡ 410 (mod 889)
x ≡ 246 (mod 289)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 739 × 889 × 289 = 189864619 - We calculate the numbers M1 to M3
M1=M/m1=189864619/739=256921,M2=M/m2=189864619/889=213571,M3=M/m3=189864619/289=656971 - 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 -53 mod 739 ≡ 686n b q r t1 t2 t3 739 256921 0 739 0 1 0 256921 739 347 488 1 0 1 739 488 1 251 0 1 -1 488 251 1 237 1 -1 2 251 237 1 14 -1 2 -3 237 14 16 13 2 -3 50 14 13 1 1 -3 50 -53 13 1 13 0 50 -53 739
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 316 mod 889 ≡ 316n b q r t1 t2 t3 889 213571 0 889 0 1 0 213571 889 240 211 1 0 1 889 211 4 45 0 1 -4 211 45 4 31 1 -4 17 45 31 1 14 -4 17 -21 31 14 2 3 17 -21 59 14 3 4 2 -21 59 -257 3 2 1 1 59 -257 316 2 1 2 0 -257 316 -889
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -82 mod 289 ≡ 207n b q r t1 t2 t3 289 656971 0 289 0 1 0 656971 289 2273 74 1 0 1 289 74 3 67 0 1 -3 74 67 1 7 1 -3 4 67 7 9 4 -3 4 -39 7 4 1 3 4 -39 43 4 3 1 1 -39 43 -82 3 1 3 0 43 -82 289
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
= (593 × 256921 × 686 +
410 × 213571 × 316 +
246 × 656971 × 207) mod 189864619
= 77537212 (mod 189864619)
So our answer is 77537212 (mod 189864619).
Verification
So we found that x ≡ 77537212
If this is correct, then the following statements (i.e. the original equations) are true:
180x (mod 739) ≡ 324 (mod 739)
824x (mod 889) ≡ 909 (mod 889)
230x (mod 289) ≡ 225 (mod 289)
Let's see whether that's indeed the case if we use x ≡ 77537212.