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 |
|---|---|---|---|---|---|---|
| 385 | 8 | 48 | 1 | 0 | 1 | -48 |
| 8 | 1 | 8 | 0 | 1 | -48 | 385 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 389 | 117 | 3 | 38 | 0 | 1 | -3 |
| 117 | 38 | 3 | 3 | 1 | -3 | 10 |
| 38 | 3 | 12 | 2 | -3 | 10 | -123 |
| 3 | 2 | 1 | 1 | 10 | -123 | 133 |
| 2 | 1 | 2 | 0 | -123 | 133 | -389 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 158 | 865 | 0 | 158 | 0 | 1 | 0 |
| 865 | 158 | 5 | 75 | 1 | 0 | 1 |
| 158 | 75 | 2 | 8 | 0 | 1 | -2 |
| 75 | 8 | 9 | 3 | 1 | -2 | 19 |
| 8 | 3 | 2 | 2 | -2 | 19 | -40 |
| 3 | 2 | 1 | 1 | 19 | -40 | 59 |
| 2 | 1 | 2 | 0 | -40 | 59 | -158 |
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 ≡ 187 × 8-1 (mod 385) ≡ 187 × 337 (mod 385) ≡ 264 (mod 385)x ≡ 330 × 117-1 (mod 389) ≡ 330 × 133 (mod 389) ≡ 322 (mod 389)x ≡ 833 × 865-1 (mod 158) ≡ 833 × 59 (mod 158) ≡ 9 (mod 158)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 264 (mod 385)
x ≡ 322 (mod 389)
x ≡ 9 (mod 158)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 385 × 389 × 158 = 23662870 - We calculate the numbers M1 to M3
M1=M/m1=23662870/385=61462,M2=M/m2=23662870/389=60830,M3=M/m3=23662870/158=149765 - 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 53 mod 385 ≡ 53n b q r t1 t2 t3 385 61462 0 385 0 1 0 61462 385 159 247 1 0 1 385 247 1 138 0 1 -1 247 138 1 109 1 -1 2 138 109 1 29 -1 2 -3 109 29 3 22 2 -3 11 29 22 1 7 -3 11 -14 22 7 3 1 11 -14 53 7 1 7 0 -14 53 -385
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 8 mod 389 ≡ 8n b q r t1 t2 t3 389 60830 0 389 0 1 0 60830 389 156 146 1 0 1 389 146 2 97 0 1 -2 146 97 1 49 1 -2 3 97 49 1 48 -2 3 -5 49 48 1 1 3 -5 8 48 1 48 0 -5 8 -389
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -25 mod 158 ≡ 133n b q r t1 t2 t3 158 149765 0 158 0 1 0 149765 158 947 139 1 0 1 158 139 1 19 0 1 -1 139 19 7 6 1 -1 8 19 6 3 1 -1 8 -25 6 1 6 0 8 -25 158
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
= (264 × 61462 × 53 +
322 × 60830 × 8 +
9 × 149765 × 133) mod 23662870
= 12799589 (mod 23662870)
So our answer is 12799589 (mod 23662870).
Verification
So we found that x ≡ 12799589
If this is correct, then the following statements (i.e. the original equations) are true:
8x (mod 385) ≡ 187 (mod 385)
117x (mod 389) ≡ 330 (mod 389)
865x (mod 158) ≡ 833 (mod 158)
Let's see whether that's indeed the case if we use x ≡ 12799589.