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.
This fills all textboxes with random numbers. If you fill in random numbers yourself, it is very likely that those numbers do not have a solution. To avoid disappointment, use this button instead! It only uses random numbers that do have a solution.
This button is similar to the "Clear everything" button, but only clears the left column.
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 |
|---|---|---|---|---|---|---|
| 321 | 860 | 0 | 321 | 0 | 1 | 0 |
| 860 | 321 | 2 | 218 | 1 | 0 | 1 |
| 321 | 218 | 1 | 103 | 0 | 1 | -1 |
| 218 | 103 | 2 | 12 | 1 | -1 | 3 |
| 103 | 12 | 8 | 7 | -1 | 3 | -25 |
| 12 | 7 | 1 | 5 | 3 | -25 | 28 |
| 7 | 5 | 1 | 2 | -25 | 28 | -53 |
| 5 | 2 | 2 | 1 | 28 | -53 | 134 |
| 2 | 1 | 2 | 0 | -53 | 134 | -321 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 563 | 752 | 0 | 563 | 0 | 1 | 0 |
| 752 | 563 | 1 | 189 | 1 | 0 | 1 |
| 563 | 189 | 2 | 185 | 0 | 1 | -2 |
| 189 | 185 | 1 | 4 | 1 | -2 | 3 |
| 185 | 4 | 46 | 1 | -2 | 3 | -140 |
| 4 | 1 | 4 | 0 | 3 | -140 | 563 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 647 | 815 | 0 | 647 | 0 | 1 | 0 |
| 815 | 647 | 1 | 168 | 1 | 0 | 1 |
| 647 | 168 | 3 | 143 | 0 | 1 | -3 |
| 168 | 143 | 1 | 25 | 1 | -3 | 4 |
| 143 | 25 | 5 | 18 | -3 | 4 | -23 |
| 25 | 18 | 1 | 7 | 4 | -23 | 27 |
| 18 | 7 | 2 | 4 | -23 | 27 | -77 |
| 7 | 4 | 1 | 3 | 27 | -77 | 104 |
| 4 | 3 | 1 | 1 | -77 | 104 | -181 |
| 3 | 1 | 3 | 0 | 104 | -181 | 647 |
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 ≡ 682 × 860-1 (mod 321) ≡ 682 × 134 (mod 321) ≡ 224 (mod 321)x ≡ 503 × 752-1 (mod 563) ≡ 503 × 423 (mod 563) ≡ 518 (mod 563)x ≡ 999 × 815-1 (mod 647) ≡ 999 × 466 (mod 647) ≡ 341 (mod 647)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 224 (mod 321)
x ≡ 518 (mod 563)
x ≡ 341 (mod 647)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 321 × 563 × 647 = 116927781 - We calculate the numbers M1 to M3
M1=M/m1=116927781/321=364261,M2=M/m2=116927781/563=207687,M3=M/m3=116927781/647=180723 - 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 13 mod 321 ≡ 13n b q r t1 t2 t3 321 364261 0 321 0 1 0 364261 321 1134 247 1 0 1 321 247 1 74 0 1 -1 247 74 3 25 1 -1 4 74 25 2 24 -1 4 -9 25 24 1 1 4 -9 13 24 1 24 0 -9 13 -321
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -122 mod 563 ≡ 441n b q r t1 t2 t3 563 207687 0 563 0 1 0 207687 563 368 503 1 0 1 563 503 1 60 0 1 -1 503 60 8 23 1 -1 9 60 23 2 14 -1 9 -19 23 14 1 9 9 -19 28 14 9 1 5 -19 28 -47 9 5 1 4 28 -47 75 5 4 1 1 -47 75 -122 4 1 4 0 75 -122 563
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 114 mod 647 ≡ 114n b q r t1 t2 t3 647 180723 0 647 0 1 0 180723 647 279 210 1 0 1 647 210 3 17 0 1 -3 210 17 12 6 1 -3 37 17 6 2 5 -3 37 -77 6 5 1 1 37 -77 114 5 1 5 0 -77 114 -647
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
= (224 × 364261 × 13 +
518 × 207687 × 441 +
341 × 180723 × 114) mod 116927781
= 105988646 (mod 116927781)
So our answer is 105988646 (mod 116927781).
Verification
So we found that x ≡ 105988646
If this is correct, then the following statements (i.e. the original equations) are true:
860x (mod 321) ≡ 682 (mod 321)
752x (mod 563) ≡ 503 (mod 563)
815x (mod 647) ≡ 999 (mod 647)
Let's see whether that's indeed the case if we use x ≡ 105988646.