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 |
|---|---|---|---|---|---|---|
| 599 | 945 | 0 | 599 | 0 | 1 | 0 |
| 945 | 599 | 1 | 346 | 1 | 0 | 1 |
| 599 | 346 | 1 | 253 | 0 | 1 | -1 |
| 346 | 253 | 1 | 93 | 1 | -1 | 2 |
| 253 | 93 | 2 | 67 | -1 | 2 | -5 |
| 93 | 67 | 1 | 26 | 2 | -5 | 7 |
| 67 | 26 | 2 | 15 | -5 | 7 | -19 |
| 26 | 15 | 1 | 11 | 7 | -19 | 26 |
| 15 | 11 | 1 | 4 | -19 | 26 | -45 |
| 11 | 4 | 2 | 3 | 26 | -45 | 116 |
| 4 | 3 | 1 | 1 | -45 | 116 | -161 |
| 3 | 1 | 3 | 0 | 116 | -161 | 599 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 677 | 891 | 0 | 677 | 0 | 1 | 0 |
| 891 | 677 | 1 | 214 | 1 | 0 | 1 |
| 677 | 214 | 3 | 35 | 0 | 1 | -3 |
| 214 | 35 | 6 | 4 | 1 | -3 | 19 |
| 35 | 4 | 8 | 3 | -3 | 19 | -155 |
| 4 | 3 | 1 | 1 | 19 | -155 | 174 |
| 3 | 1 | 3 | 0 | -155 | 174 | -677 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 233 | 457 | 0 | 233 | 0 | 1 | 0 |
| 457 | 233 | 1 | 224 | 1 | 0 | 1 |
| 233 | 224 | 1 | 9 | 0 | 1 | -1 |
| 224 | 9 | 24 | 8 | 1 | -1 | 25 |
| 9 | 8 | 1 | 1 | -1 | 25 | -26 |
| 8 | 1 | 8 | 0 | 25 | -26 | 233 |
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 ≡ 979 × 945-1 (mod 599) ≡ 979 × 438 (mod 599) ≡ 517 (mod 599)x ≡ 546 × 891-1 (mod 677) ≡ 546 × 174 (mod 677) ≡ 224 (mod 677)x ≡ 381 × 457-1 (mod 233) ≡ 381 × 207 (mod 233) ≡ 113 (mod 233)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 517 (mod 599)
x ≡ 224 (mod 677)
x ≡ 113 (mod 233)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 599 × 677 × 233 = 94486859 - We calculate the numbers M1 to M3
M1=M/m1=94486859/599=157741,M2=M/m2=94486859/677=139567,M3=M/m3=94486859/233=405523 - 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 -138 mod 599 ≡ 461n b q r t1 t2 t3 599 157741 0 599 0 1 0 157741 599 263 204 1 0 1 599 204 2 191 0 1 -2 204 191 1 13 1 -2 3 191 13 14 9 -2 3 -44 13 9 1 4 3 -44 47 9 4 2 1 -44 47 -138 4 1 4 0 47 -138 599
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -245 mod 677 ≡ 432n b q r t1 t2 t3 677 139567 0 677 0 1 0 139567 677 206 105 1 0 1 677 105 6 47 0 1 -6 105 47 2 11 1 -6 13 47 11 4 3 -6 13 -58 11 3 3 2 13 -58 187 3 2 1 1 -58 187 -245 2 1 2 0 187 -245 677
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -95 mod 233 ≡ 138n b q r t1 t2 t3 233 405523 0 233 0 1 0 405523 233 1740 103 1 0 1 233 103 2 27 0 1 -2 103 27 3 22 1 -2 7 27 22 1 5 -2 7 -9 22 5 4 2 7 -9 43 5 2 2 1 -9 43 -95 2 1 2 0 43 -95 233
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
= (517 × 157741 × 461 +
224 × 139567 × 432 +
113 × 405523 × 138) mod 94486859
= 71338422 (mod 94486859)
So our answer is 71338422 (mod 94486859).
Verification
So we found that x ≡ 71338422
If this is correct, then the following statements (i.e. the original equations) are true:
945x (mod 599) ≡ 979 (mod 599)
891x (mod 677) ≡ 546 (mod 677)
457x (mod 233) ≡ 381 (mod 233)
Let's see whether that's indeed the case if we use x ≡ 71338422.