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 |
|---|---|---|---|---|---|---|
| 829 | 407 | 2 | 15 | 0 | 1 | -2 |
| 407 | 15 | 27 | 2 | 1 | -2 | 55 |
| 15 | 2 | 7 | 1 | -2 | 55 | -387 |
| 2 | 1 | 2 | 0 | 55 | -387 | 829 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 668 | 861 | 0 | 668 | 0 | 1 | 0 |
| 861 | 668 | 1 | 193 | 1 | 0 | 1 |
| 668 | 193 | 3 | 89 | 0 | 1 | -3 |
| 193 | 89 | 2 | 15 | 1 | -3 | 7 |
| 89 | 15 | 5 | 14 | -3 | 7 | -38 |
| 15 | 14 | 1 | 1 | 7 | -38 | 45 |
| 14 | 1 | 14 | 0 | -38 | 45 | -668 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 109 | 566 | 0 | 109 | 0 | 1 | 0 |
| 566 | 109 | 5 | 21 | 1 | 0 | 1 |
| 109 | 21 | 5 | 4 | 0 | 1 | -5 |
| 21 | 4 | 5 | 1 | 1 | -5 | 26 |
| 4 | 1 | 4 | 0 | -5 | 26 | -109 |
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 ≡ 720 × 407-1 (mod 829) ≡ 720 × 442 (mod 829) ≡ 733 (mod 829)x ≡ 737 × 861-1 (mod 668) ≡ 737 × 45 (mod 668) ≡ 433 (mod 668)x ≡ 660 × 566-1 (mod 109) ≡ 660 × 26 (mod 109) ≡ 47 (mod 109)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 733 (mod 829)
x ≡ 433 (mod 668)
x ≡ 47 (mod 109)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 829 × 668 × 109 = 60361148 - We calculate the numbers M1 to M3
M1=M/m1=60361148/829=72812,M2=M/m2=60361148/668=90361,M3=M/m3=60361148/109=553772 - 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 -302 mod 829 ≡ 527n b q r t1 t2 t3 829 72812 0 829 0 1 0 72812 829 87 689 1 0 1 829 689 1 140 0 1 -1 689 140 4 129 1 -1 5 140 129 1 11 -1 5 -6 129 11 11 8 5 -6 71 11 8 1 3 -6 71 -77 8 3 2 2 71 -77 225 3 2 1 1 -77 225 -302 2 1 2 0 225 -302 829
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -155 mod 668 ≡ 513n b q r t1 t2 t3 668 90361 0 668 0 1 0 90361 668 135 181 1 0 1 668 181 3 125 0 1 -3 181 125 1 56 1 -3 4 125 56 2 13 -3 4 -11 56 13 4 4 4 -11 48 13 4 3 1 -11 48 -155 4 1 4 0 48 -155 668
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -44 mod 109 ≡ 65n b q r t1 t2 t3 109 553772 0 109 0 1 0 553772 109 5080 52 1 0 1 109 52 2 5 0 1 -2 52 5 10 2 1 -2 21 5 2 2 1 -2 21 -44 2 1 2 0 21 -44 109
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
= (733 × 72812 × 527 +
433 × 90361 × 513 +
47 × 553772 × 65) mod 60361148
= 31884073 (mod 60361148)
So our answer is 31884073 (mod 60361148).
Verification
So we found that x ≡ 31884073
If this is correct, then the following statements (i.e. the original equations) are true:
407x (mod 829) ≡ 720 (mod 829)
861x (mod 668) ≡ 737 (mod 668)
566x (mod 109) ≡ 660 (mod 109)
Let's see whether that's indeed the case if we use x ≡ 31884073.