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 |
|---|---|---|---|---|---|---|
| 397 | 176 | 2 | 45 | 0 | 1 | -2 |
| 176 | 45 | 3 | 41 | 1 | -2 | 7 |
| 45 | 41 | 1 | 4 | -2 | 7 | -9 |
| 41 | 4 | 10 | 1 | 7 | -9 | 97 |
| 4 | 1 | 4 | 0 | -9 | 97 | -397 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 254 | 447 | 0 | 254 | 0 | 1 | 0 |
| 447 | 254 | 1 | 193 | 1 | 0 | 1 |
| 254 | 193 | 1 | 61 | 0 | 1 | -1 |
| 193 | 61 | 3 | 10 | 1 | -1 | 4 |
| 61 | 10 | 6 | 1 | -1 | 4 | -25 |
| 10 | 1 | 10 | 0 | 4 | -25 | 254 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 415 | 762 | 0 | 415 | 0 | 1 | 0 |
| 762 | 415 | 1 | 347 | 1 | 0 | 1 |
| 415 | 347 | 1 | 68 | 0 | 1 | -1 |
| 347 | 68 | 5 | 7 | 1 | -1 | 6 |
| 68 | 7 | 9 | 5 | -1 | 6 | -55 |
| 7 | 5 | 1 | 2 | 6 | -55 | 61 |
| 5 | 2 | 2 | 1 | -55 | 61 | -177 |
| 2 | 1 | 2 | 0 | 61 | -177 | 415 |
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 ≡ 96 × 176-1 (mod 397) ≡ 96 × 97 (mod 397) ≡ 181 (mod 397)x ≡ 768 × 447-1 (mod 254) ≡ 768 × 229 (mod 254) ≡ 104 (mod 254)x ≡ 266 × 762-1 (mod 415) ≡ 266 × 238 (mod 415) ≡ 228 (mod 415)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 181 (mod 397)
x ≡ 104 (mod 254)
x ≡ 228 (mod 415)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 397 × 254 × 415 = 41847770 - We calculate the numbers M1 to M3
M1=M/m1=41847770/397=105410,M2=M/m2=41847770/254=164755,M3=M/m3=41847770/415=100838 - 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 -122 mod 397 ≡ 275n b q r t1 t2 t3 397 105410 0 397 0 1 0 105410 397 265 205 1 0 1 397 205 1 192 0 1 -1 205 192 1 13 1 -1 2 192 13 14 10 -1 2 -29 13 10 1 3 2 -29 31 10 3 3 1 -29 31 -122 3 1 3 0 31 -122 397
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -67 mod 254 ≡ 187n b q r t1 t2 t3 254 164755 0 254 0 1 0 164755 254 648 163 1 0 1 254 163 1 91 0 1 -1 163 91 1 72 1 -1 2 91 72 1 19 -1 2 -3 72 19 3 15 2 -3 11 19 15 1 4 -3 11 -14 15 4 3 3 11 -14 53 4 3 1 1 -14 53 -67 3 1 3 0 53 -67 254
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -178 mod 415 ≡ 237n b q r t1 t2 t3 415 100838 0 415 0 1 0 100838 415 242 408 1 0 1 415 408 1 7 0 1 -1 408 7 58 2 1 -1 59 7 2 3 1 -1 59 -178 2 1 2 0 59 -178 415
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
= (181 × 105410 × 275 +
104 × 164755 × 187 +
228 × 100838 × 237) mod 41847770
= 6360518 (mod 41847770)
So our answer is 6360518 (mod 41847770).
Verification
So we found that x ≡ 6360518
If this is correct, then the following statements (i.e. the original equations) are true:
176x (mod 397) ≡ 96 (mod 397)
447x (mod 254) ≡ 768 (mod 254)
762x (mod 415) ≡ 266 (mod 415)
Let's see whether that's indeed the case if we use x ≡ 6360518.