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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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 |
|---|---|---|---|---|---|---|
| 815 | 33 | 24 | 23 | 0 | 1 | -24 |
| 33 | 23 | 1 | 10 | 1 | -24 | 25 |
| 23 | 10 | 2 | 3 | -24 | 25 | -74 |
| 10 | 3 | 3 | 1 | 25 | -74 | 247 |
| 3 | 1 | 3 | 0 | -74 | 247 | -815 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 611 | 206 | 2 | 199 | 0 | 1 | -2 |
| 206 | 199 | 1 | 7 | 1 | -2 | 3 |
| 199 | 7 | 28 | 3 | -2 | 3 | -86 |
| 7 | 3 | 2 | 1 | 3 | -86 | 175 |
| 3 | 1 | 3 | 0 | -86 | 175 | -611 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 89 | 4 | 22 | 1 | 0 | 1 | -22 |
| 4 | 1 | 4 | 0 | 1 | -22 | 89 |
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 ≡ 931 × 33-1 (mod 815) ≡ 931 × 247 (mod 815) ≡ 127 (mod 815)x ≡ 287 × 206-1 (mod 611) ≡ 287 × 175 (mod 611) ≡ 123 (mod 611)x ≡ 477 × 4-1 (mod 89) ≡ 477 × 67 (mod 89) ≡ 8 (mod 89)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 127 (mod 815)
x ≡ 123 (mod 611)
x ≡ 8 (mod 89)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 815 × 611 × 89 = 44318885 - We calculate the numbers M1 to M3
M1=M/m1=44318885/815=54379,M2=M/m2=44318885/611=72535,M3=M/m3=44318885/89=497965 - 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 119 mod 815 ≡ 119n b q r t1 t2 t3 815 54379 0 815 0 1 0 54379 815 66 589 1 0 1 815 589 1 226 0 1 -1 589 226 2 137 1 -1 3 226 137 1 89 -1 3 -4 137 89 1 48 3 -4 7 89 48 1 41 -4 7 -11 48 41 1 7 7 -11 18 41 7 5 6 -11 18 -101 7 6 1 1 18 -101 119 6 1 6 0 -101 119 -815
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -151 mod 611 ≡ 460n b q r t1 t2 t3 611 72535 0 611 0 1 0 72535 611 118 437 1 0 1 611 437 1 174 0 1 -1 437 174 2 89 1 -1 3 174 89 1 85 -1 3 -4 89 85 1 4 3 -4 7 85 4 21 1 -4 7 -151 4 1 4 0 7 -151 611
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 9 mod 89 ≡ 9n b q r t1 t2 t3 89 497965 0 89 0 1 0 497965 89 5595 10 1 0 1 89 10 8 9 0 1 -8 10 9 1 1 1 -8 9 9 1 9 0 -8 9 -89
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
= (127 × 54379 × 119 +
123 × 72535 × 460 +
8 × 497965 × 9) mod 44318885
= 42317372 (mod 44318885)
So our answer is 42317372 (mod 44318885).
Verification
So we found that x ≡ 42317372
If this is correct, then the following statements (i.e. the original equations) are true:
33x (mod 815) ≡ 931 (mod 815)
206x (mod 611) ≡ 287 (mod 611)
4x (mod 89) ≡ 477 (mod 89)
Let's see whether that's indeed the case if we use x ≡ 42317372.