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 |
|---|---|---|---|---|---|---|
| 991 | 229 | 4 | 75 | 0 | 1 | -4 |
| 229 | 75 | 3 | 4 | 1 | -4 | 13 |
| 75 | 4 | 18 | 3 | -4 | 13 | -238 |
| 4 | 3 | 1 | 1 | 13 | -238 | 251 |
| 3 | 1 | 3 | 0 | -238 | 251 | -991 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 617 | 130 | 4 | 97 | 0 | 1 | -4 |
| 130 | 97 | 1 | 33 | 1 | -4 | 5 |
| 97 | 33 | 2 | 31 | -4 | 5 | -14 |
| 33 | 31 | 1 | 2 | 5 | -14 | 19 |
| 31 | 2 | 15 | 1 | -14 | 19 | -299 |
| 2 | 1 | 2 | 0 | 19 | -299 | 617 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 362 | 449 | 0 | 362 | 0 | 1 | 0 |
| 449 | 362 | 1 | 87 | 1 | 0 | 1 |
| 362 | 87 | 4 | 14 | 0 | 1 | -4 |
| 87 | 14 | 6 | 3 | 1 | -4 | 25 |
| 14 | 3 | 4 | 2 | -4 | 25 | -104 |
| 3 | 2 | 1 | 1 | 25 | -104 | 129 |
| 2 | 1 | 2 | 0 | -104 | 129 | -362 |
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 ≡ 449 × 229-1 (mod 991) ≡ 449 × 251 (mod 991) ≡ 716 (mod 991)x ≡ 288 × 130-1 (mod 617) ≡ 288 × 318 (mod 617) ≡ 268 (mod 617)x ≡ 689 × 449-1 (mod 362) ≡ 689 × 129 (mod 362) ≡ 191 (mod 362)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 716 (mod 991)
x ≡ 268 (mod 617)
x ≡ 191 (mod 362)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 991 × 617 × 362 = 221343814 - We calculate the numbers M1 to M3
M1=M/m1=221343814/991=223354,M2=M/m2=221343814/617=358742,M3=M/m3=221343814/362=611447 - 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 -319 mod 991 ≡ 672n b q r t1 t2 t3 991 223354 0 991 0 1 0 223354 991 225 379 1 0 1 991 379 2 233 0 1 -2 379 233 1 146 1 -2 3 233 146 1 87 -2 3 -5 146 87 1 59 3 -5 8 87 59 1 28 -5 8 -13 59 28 2 3 8 -13 34 28 3 9 1 -13 34 -319 3 1 3 0 34 -319 991
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 156 mod 617 ≡ 156n b q r t1 t2 t3 617 358742 0 617 0 1 0 358742 617 581 265 1 0 1 617 265 2 87 0 1 -2 265 87 3 4 1 -2 7 87 4 21 3 -2 7 -149 4 3 1 1 7 -149 156 3 1 3 0 -149 156 -617
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 25 mod 362 ≡ 25n b q r t1 t2 t3 362 611447 0 362 0 1 0 611447 362 1689 29 1 0 1 362 29 12 14 0 1 -12 29 14 2 1 1 -12 25 14 1 14 0 -12 25 -362
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
= (716 × 223354 × 672 +
268 × 358742 × 156 +
191 × 611447 × 25) mod 221343814
= 104570045 (mod 221343814)
So our answer is 104570045 (mod 221343814).
Verification
So we found that x ≡ 104570045
If this is correct, then the following statements (i.e. the original equations) are true:
229x (mod 991) ≡ 449 (mod 991)
130x (mod 617) ≡ 288 (mod 617)
449x (mod 362) ≡ 689 (mod 362)
Let's see whether that's indeed the case if we use x ≡ 104570045.