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 |
|---|---|---|---|---|---|---|
| 353 | 730 | 0 | 353 | 0 | 1 | 0 |
| 730 | 353 | 2 | 24 | 1 | 0 | 1 |
| 353 | 24 | 14 | 17 | 0 | 1 | -14 |
| 24 | 17 | 1 | 7 | 1 | -14 | 15 |
| 17 | 7 | 2 | 3 | -14 | 15 | -44 |
| 7 | 3 | 2 | 1 | 15 | -44 | 103 |
| 3 | 1 | 3 | 0 | -44 | 103 | -353 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 259 | 415 | 0 | 259 | 0 | 1 | 0 |
| 415 | 259 | 1 | 156 | 1 | 0 | 1 |
| 259 | 156 | 1 | 103 | 0 | 1 | -1 |
| 156 | 103 | 1 | 53 | 1 | -1 | 2 |
| 103 | 53 | 1 | 50 | -1 | 2 | -3 |
| 53 | 50 | 1 | 3 | 2 | -3 | 5 |
| 50 | 3 | 16 | 2 | -3 | 5 | -83 |
| 3 | 2 | 1 | 1 | 5 | -83 | 88 |
| 2 | 1 | 2 | 0 | -83 | 88 | -259 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 417 | 812 | 0 | 417 | 0 | 1 | 0 |
| 812 | 417 | 1 | 395 | 1 | 0 | 1 |
| 417 | 395 | 1 | 22 | 0 | 1 | -1 |
| 395 | 22 | 17 | 21 | 1 | -1 | 18 |
| 22 | 21 | 1 | 1 | -1 | 18 | -19 |
| 21 | 1 | 21 | 0 | 18 | -19 | 417 |
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 ≡ 93 × 730-1 (mod 353) ≡ 93 × 103 (mod 353) ≡ 48 (mod 353)x ≡ 179 × 415-1 (mod 259) ≡ 179 × 88 (mod 259) ≡ 212 (mod 259)x ≡ 419 × 812-1 (mod 417) ≡ 419 × 398 (mod 417) ≡ 379 (mod 417)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 48 (mod 353)
x ≡ 212 (mod 259)
x ≡ 379 (mod 417)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 353 × 259 × 417 = 38125059 - We calculate the numbers M1 to M3
M1=M/m1=38125059/353=108003,M2=M/m2=38125059/259=147201,M3=M/m3=38125059/417=91427 - 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 47 mod 353 ≡ 47n b q r t1 t2 t3 353 108003 0 353 0 1 0 108003 353 305 338 1 0 1 353 338 1 15 0 1 -1 338 15 22 8 1 -1 23 15 8 1 7 -1 23 -24 8 7 1 1 23 -24 47 7 1 7 0 -24 47 -353
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -32 mod 259 ≡ 227n b q r t1 t2 t3 259 147201 0 259 0 1 0 147201 259 568 89 1 0 1 259 89 2 81 0 1 -2 89 81 1 8 1 -2 3 81 8 10 1 -2 3 -32 8 1 8 0 3 -32 259
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -4 mod 417 ≡ 413n b q r t1 t2 t3 417 91427 0 417 0 1 0 91427 417 219 104 1 0 1 417 104 4 1 0 1 -4 104 1 104 0 1 -4 417
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
= (48 × 108003 × 47 +
212 × 147201 × 227 +
379 × 91427 × 413) mod 38125059
= 21441268 (mod 38125059)
So our answer is 21441268 (mod 38125059).
Verification
So we found that x ≡ 21441268
If this is correct, then the following statements (i.e. the original equations) are true:
730x (mod 353) ≡ 93 (mod 353)
415x (mod 259) ≡ 179 (mod 259)
812x (mod 417) ≡ 419 (mod 417)
Let's see whether that's indeed the case if we use x ≡ 21441268.