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 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 |
|---|---|---|---|---|---|---|
| 199 | 810 | 0 | 199 | 0 | 1 | 0 |
| 810 | 199 | 4 | 14 | 1 | 0 | 1 |
| 199 | 14 | 14 | 3 | 0 | 1 | -14 |
| 14 | 3 | 4 | 2 | 1 | -14 | 57 |
| 3 | 2 | 1 | 1 | -14 | 57 | -71 |
| 2 | 1 | 2 | 0 | 57 | -71 | 199 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 997 | 648 | 1 | 349 | 0 | 1 | -1 |
| 648 | 349 | 1 | 299 | 1 | -1 | 2 |
| 349 | 299 | 1 | 50 | -1 | 2 | -3 |
| 299 | 50 | 5 | 49 | 2 | -3 | 17 |
| 50 | 49 | 1 | 1 | -3 | 17 | -20 |
| 49 | 1 | 49 | 0 | 17 | -20 | 997 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 229 | 5 | 45 | 4 | 0 | 1 | -45 |
| 5 | 4 | 1 | 1 | 1 | -45 | 46 |
| 4 | 1 | 4 | 0 | -45 | 46 | -229 |
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 ≡ 951 × 810-1 (mod 199) ≡ 951 × 128 (mod 199) ≡ 139 (mod 199)x ≡ 247 × 648-1 (mod 997) ≡ 247 × 977 (mod 997) ≡ 45 (mod 997)x ≡ 480 × 5-1 (mod 229) ≡ 480 × 46 (mod 229) ≡ 96 (mod 229)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 139 (mod 199)
x ≡ 45 (mod 997)
x ≡ 96 (mod 229)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 199 × 997 × 229 = 45434287 - We calculate the numbers M1 to M3
M1=M/m1=45434287/199=228313,M2=M/m2=45434287/997=45571,M3=M/m3=45434287/229=198403 - 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 -63 mod 199 ≡ 136n b q r t1 t2 t3 199 228313 0 199 0 1 0 228313 199 1147 60 1 0 1 199 60 3 19 0 1 -3 60 19 3 3 1 -3 10 19 3 6 1 -3 10 -63 3 1 3 0 10 -63 199
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 394 mod 997 ≡ 394n b q r t1 t2 t3 997 45571 0 997 0 1 0 45571 997 45 706 1 0 1 997 706 1 291 0 1 -1 706 291 2 124 1 -1 3 291 124 2 43 -1 3 -7 124 43 2 38 3 -7 17 43 38 1 5 -7 17 -24 38 5 7 3 17 -24 185 5 3 1 2 -24 185 -209 3 2 1 1 185 -209 394 2 1 2 0 -209 394 -997
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -18 mod 229 ≡ 211n b q r t1 t2 t3 229 198403 0 229 0 1 0 198403 229 866 89 1 0 1 229 89 2 51 0 1 -2 89 51 1 38 1 -2 3 51 38 1 13 -2 3 -5 38 13 2 12 3 -5 13 13 12 1 1 -5 13 -18 12 1 12 0 13 -18 229
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
= (139 × 228313 × 136 +
45 × 45571 × 394 +
96 × 198403 × 211) mod 45434287
= 10562263 (mod 45434287)
So our answer is 10562263 (mod 45434287).
Verification
So we found that x ≡ 10562263
If this is correct, then the following statements (i.e. the original equations) are true:
810x (mod 199) ≡ 951 (mod 199)
648x (mod 997) ≡ 247 (mod 997)
5x (mod 229) ≡ 480 (mod 229)
Let's see whether that's indeed the case if we use x ≡ 10562263.