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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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 |
|---|---|---|---|---|---|---|
| 991 | 342 | 2 | 307 | 0 | 1 | -2 |
| 342 | 307 | 1 | 35 | 1 | -2 | 3 |
| 307 | 35 | 8 | 27 | -2 | 3 | -26 |
| 35 | 27 | 1 | 8 | 3 | -26 | 29 |
| 27 | 8 | 3 | 3 | -26 | 29 | -113 |
| 8 | 3 | 2 | 2 | 29 | -113 | 255 |
| 3 | 2 | 1 | 1 | -113 | 255 | -368 |
| 2 | 1 | 2 | 0 | 255 | -368 | 991 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 949 | 106 | 8 | 101 | 0 | 1 | -8 |
| 106 | 101 | 1 | 5 | 1 | -8 | 9 |
| 101 | 5 | 20 | 1 | -8 | 9 | -188 |
| 5 | 1 | 5 | 0 | 9 | -188 | 949 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 877 | 153 | 5 | 112 | 0 | 1 | -5 |
| 153 | 112 | 1 | 41 | 1 | -5 | 6 |
| 112 | 41 | 2 | 30 | -5 | 6 | -17 |
| 41 | 30 | 1 | 11 | 6 | -17 | 23 |
| 30 | 11 | 2 | 8 | -17 | 23 | -63 |
| 11 | 8 | 1 | 3 | 23 | -63 | 86 |
| 8 | 3 | 2 | 2 | -63 | 86 | -235 |
| 3 | 2 | 1 | 1 | 86 | -235 | 321 |
| 2 | 1 | 2 | 0 | -235 | 321 | -877 |
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 ≡ 506 × 342-1 (mod 991) ≡ 506 × 623 (mod 991) ≡ 100 (mod 991)x ≡ 207 × 106-1 (mod 949) ≡ 207 × 761 (mod 949) ≡ 942 (mod 949)x ≡ 906 × 153-1 (mod 877) ≡ 906 × 321 (mod 877) ≡ 539 (mod 877)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 100 (mod 991)
x ≡ 942 (mod 949)
x ≡ 539 (mod 877)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 991 × 949 × 877 = 824782543 - We calculate the numbers M1 to M3
M1=M/m1=824782543/991=832273,M2=M/m2=824782543/949=869107,M3=M/m3=824782543/877=940459 - 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 -451 mod 991 ≡ 540n b q r t1 t2 t3 991 832273 0 991 0 1 0 832273 991 839 824 1 0 1 991 824 1 167 0 1 -1 824 167 4 156 1 -1 5 167 156 1 11 -1 5 -6 156 11 14 2 5 -6 89 11 2 5 1 -6 89 -451 2 1 2 0 89 -451 991
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -252 mod 949 ≡ 697n b q r t1 t2 t3 949 869107 0 949 0 1 0 869107 949 915 772 1 0 1 949 772 1 177 0 1 -1 772 177 4 64 1 -1 5 177 64 2 49 -1 5 -11 64 49 1 15 5 -11 16 49 15 3 4 -11 16 -59 15 4 3 3 16 -59 193 4 3 1 1 -59 193 -252 3 1 3 0 193 -252 949
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -245 mod 877 ≡ 632n b q r t1 t2 t3 877 940459 0 877 0 1 0 940459 877 1072 315 1 0 1 877 315 2 247 0 1 -2 315 247 1 68 1 -2 3 247 68 3 43 -2 3 -11 68 43 1 25 3 -11 14 43 25 1 18 -11 14 -25 25 18 1 7 14 -25 39 18 7 2 4 -25 39 -103 7 4 1 3 39 -103 142 4 3 1 1 -103 142 -245 3 1 3 0 142 -245 877
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
= (100 × 832273 × 540 +
942 × 869107 × 697 +
539 × 940459 × 632) mod 824782543
= 637875088 (mod 824782543)
So our answer is 637875088 (mod 824782543).
Verification
So we found that x ≡ 637875088
If this is correct, then the following statements (i.e. the original equations) are true:
342x (mod 991) ≡ 506 (mod 991)
106x (mod 949) ≡ 207 (mod 949)
153x (mod 877) ≡ 906 (mod 877)
Let's see whether that's indeed the case if we use x ≡ 637875088.