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 |
|---|---|---|---|---|---|---|
| 739 | 958 | 0 | 739 | 0 | 1 | 0 |
| 958 | 739 | 1 | 219 | 1 | 0 | 1 |
| 739 | 219 | 3 | 82 | 0 | 1 | -3 |
| 219 | 82 | 2 | 55 | 1 | -3 | 7 |
| 82 | 55 | 1 | 27 | -3 | 7 | -10 |
| 55 | 27 | 2 | 1 | 7 | -10 | 27 |
| 27 | 1 | 27 | 0 | -10 | 27 | -739 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 217 | 890 | 0 | 217 | 0 | 1 | 0 |
| 890 | 217 | 4 | 22 | 1 | 0 | 1 |
| 217 | 22 | 9 | 19 | 0 | 1 | -9 |
| 22 | 19 | 1 | 3 | 1 | -9 | 10 |
| 19 | 3 | 6 | 1 | -9 | 10 | -69 |
| 3 | 1 | 3 | 0 | 10 | -69 | 217 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 643 | 30 | 21 | 13 | 0 | 1 | -21 |
| 30 | 13 | 2 | 4 | 1 | -21 | 43 |
| 13 | 4 | 3 | 1 | -21 | 43 | -150 |
| 4 | 1 | 4 | 0 | 43 | -150 | 643 |
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 ≡ 173 × 958-1 (mod 739) ≡ 173 × 27 (mod 739) ≡ 237 (mod 739)x ≡ 639 × 890-1 (mod 217) ≡ 639 × 148 (mod 217) ≡ 177 (mod 217)x ≡ 809 × 30-1 (mod 643) ≡ 809 × 493 (mod 643) ≡ 177 (mod 643)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 237 (mod 739)
x ≡ 177 (mod 217)
x ≡ 177 (mod 643)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 739 × 217 × 643 = 103113409 - We calculate the numbers M1 to M3
M1=M/m1=103113409/739=139531,M2=M/m2=103113409/217=475177,M3=M/m3=103113409/643=160363 - 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 -322 mod 739 ≡ 417n b q r t1 t2 t3 739 139531 0 739 0 1 0 139531 739 188 599 1 0 1 739 599 1 140 0 1 -1 599 140 4 39 1 -1 5 140 39 3 23 -1 5 -16 39 23 1 16 5 -16 21 23 16 1 7 -16 21 -37 16 7 2 2 21 -37 95 7 2 3 1 -37 95 -322 2 1 2 0 95 -322 739
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -86 mod 217 ≡ 131n b q r t1 t2 t3 217 475177 0 217 0 1 0 475177 217 2189 164 1 0 1 217 164 1 53 0 1 -1 164 53 3 5 1 -1 4 53 5 10 3 -1 4 -41 5 3 1 2 4 -41 45 3 2 1 1 -41 45 -86 2 1 2 0 45 -86 217
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -108 mod 643 ≡ 535n b q r t1 t2 t3 643 160363 0 643 0 1 0 160363 643 249 256 1 0 1 643 256 2 131 0 1 -2 256 131 1 125 1 -2 3 131 125 1 6 -2 3 -5 125 6 20 5 3 -5 103 6 5 1 1 -5 103 -108 5 1 5 0 103 -108 643
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
= (237 × 139531 × 417 +
177 × 475177 × 131 +
177 × 160363 × 535) mod 103113409
= 88323300 (mod 103113409)
So our answer is 88323300 (mod 103113409).
Verification
So we found that x ≡ 88323300
If this is correct, then the following statements (i.e. the original equations) are true:
958x (mod 739) ≡ 173 (mod 739)
890x (mod 217) ≡ 639 (mod 217)
30x (mod 643) ≡ 809 (mod 643)
Let's see whether that's indeed the case if we use x ≡ 88323300.