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 |
|---|---|---|---|---|---|---|
| 603 | 416 | 1 | 187 | 0 | 1 | -1 |
| 416 | 187 | 2 | 42 | 1 | -1 | 3 |
| 187 | 42 | 4 | 19 | -1 | 3 | -13 |
| 42 | 19 | 2 | 4 | 3 | -13 | 29 |
| 19 | 4 | 4 | 3 | -13 | 29 | -129 |
| 4 | 3 | 1 | 1 | 29 | -129 | 158 |
| 3 | 1 | 3 | 0 | -129 | 158 | -603 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 353 | 708 | 0 | 353 | 0 | 1 | 0 |
| 708 | 353 | 2 | 2 | 1 | 0 | 1 |
| 353 | 2 | 176 | 1 | 0 | 1 | -176 |
| 2 | 1 | 2 | 0 | 1 | -176 | 353 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 689 | 615 | 1 | 74 | 0 | 1 | -1 |
| 615 | 74 | 8 | 23 | 1 | -1 | 9 |
| 74 | 23 | 3 | 5 | -1 | 9 | -28 |
| 23 | 5 | 4 | 3 | 9 | -28 | 121 |
| 5 | 3 | 1 | 2 | -28 | 121 | -149 |
| 3 | 2 | 1 | 1 | 121 | -149 | 270 |
| 2 | 1 | 2 | 0 | -149 | 270 | -689 |
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 ≡ 608 × 416-1 (mod 603) ≡ 608 × 158 (mod 603) ≡ 187 (mod 603)x ≡ 269 × 708-1 (mod 353) ≡ 269 × 177 (mod 353) ≡ 311 (mod 353)x ≡ 994 × 615-1 (mod 689) ≡ 994 × 270 (mod 689) ≡ 359 (mod 689)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 187 (mod 603)
x ≡ 311 (mod 353)
x ≡ 359 (mod 689)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 603 × 353 × 689 = 146659851 - We calculate the numbers M1 to M3
M1=M/m1=146659851/603=243217,M2=M/m2=146659851/353=415467,M3=M/m3=146659851/689=212859 - 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 -287 mod 603 ≡ 316n b q r t1 t2 t3 603 243217 0 603 0 1 0 243217 603 403 208 1 0 1 603 208 2 187 0 1 -2 208 187 1 21 1 -2 3 187 21 8 19 -2 3 -26 21 19 1 2 3 -26 29 19 2 9 1 -26 29 -287 2 1 2 0 29 -287 603
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 126 mod 353 ≡ 126n b q r t1 t2 t3 353 415467 0 353 0 1 0 415467 353 1176 339 1 0 1 353 339 1 14 0 1 -1 339 14 24 3 1 -1 25 14 3 4 2 -1 25 -101 3 2 1 1 25 -101 126 2 1 2 0 -101 126 -353
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 82 mod 689 ≡ 82n b q r t1 t2 t3 689 212859 0 689 0 1 0 212859 689 308 647 1 0 1 689 647 1 42 0 1 -1 647 42 15 17 1 -1 16 42 17 2 8 -1 16 -33 17 8 2 1 16 -33 82 8 1 8 0 -33 82 -689
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
= (187 × 243217 × 316 +
311 × 415467 × 126 +
359 × 212859 × 82) mod 146659851
= 107189467 (mod 146659851)
So our answer is 107189467 (mod 146659851).
Verification
So we found that x ≡ 107189467
If this is correct, then the following statements (i.e. the original equations) are true:
416x (mod 603) ≡ 608 (mod 603)
708x (mod 353) ≡ 269 (mod 353)
615x (mod 689) ≡ 994 (mod 689)
Let's see whether that's indeed the case if we use x ≡ 107189467.