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 |
|---|---|---|---|---|---|---|
| 61 | 576 | 0 | 61 | 0 | 1 | 0 |
| 576 | 61 | 9 | 27 | 1 | 0 | 1 |
| 61 | 27 | 2 | 7 | 0 | 1 | -2 |
| 27 | 7 | 3 | 6 | 1 | -2 | 7 |
| 7 | 6 | 1 | 1 | -2 | 7 | -9 |
| 6 | 1 | 6 | 0 | 7 | -9 | 61 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 711 | 449 | 1 | 262 | 0 | 1 | -1 |
| 449 | 262 | 1 | 187 | 1 | -1 | 2 |
| 262 | 187 | 1 | 75 | -1 | 2 | -3 |
| 187 | 75 | 2 | 37 | 2 | -3 | 8 |
| 75 | 37 | 2 | 1 | -3 | 8 | -19 |
| 37 | 1 | 37 | 0 | 8 | -19 | 711 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 881 | 797 | 1 | 84 | 0 | 1 | -1 |
| 797 | 84 | 9 | 41 | 1 | -1 | 10 |
| 84 | 41 | 2 | 2 | -1 | 10 | -21 |
| 41 | 2 | 20 | 1 | 10 | -21 | 430 |
| 2 | 1 | 2 | 0 | -21 | 430 | -881 |
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 ≡ 560 × 576-1 (mod 61) ≡ 560 × 52 (mod 61) ≡ 23 (mod 61)x ≡ 645 × 449-1 (mod 711) ≡ 645 × 692 (mod 711) ≡ 543 (mod 711)x ≡ 714 × 797-1 (mod 881) ≡ 714 × 430 (mod 881) ≡ 432 (mod 881)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 23 (mod 61)
x ≡ 543 (mod 711)
x ≡ 432 (mod 881)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 61 × 711 × 881 = 38209851 - We calculate the numbers M1 to M3
M1=M/m1=38209851/61=626391,M2=M/m2=38209851/711=53741,M3=M/m3=38209851/881=43371 - 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 -17 mod 61 ≡ 44n b q r t1 t2 t3 61 626391 0 61 0 1 0 626391 61 10268 43 1 0 1 61 43 1 18 0 1 -1 43 18 2 7 1 -1 3 18 7 2 4 -1 3 -7 7 4 1 3 3 -7 10 4 3 1 1 -7 10 -17 3 1 3 0 10 -17 61
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -94 mod 711 ≡ 617n b q r t1 t2 t3 711 53741 0 711 0 1 0 53741 711 75 416 1 0 1 711 416 1 295 0 1 -1 416 295 1 121 1 -1 2 295 121 2 53 -1 2 -5 121 53 2 15 2 -5 12 53 15 3 8 -5 12 -41 15 8 1 7 12 -41 53 8 7 1 1 -41 53 -94 7 1 7 0 53 -94 711
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 362 mod 881 ≡ 362n b q r t1 t2 t3 881 43371 0 881 0 1 0 43371 881 49 202 1 0 1 881 202 4 73 0 1 -4 202 73 2 56 1 -4 9 73 56 1 17 -4 9 -13 56 17 3 5 9 -13 48 17 5 3 2 -13 48 -157 5 2 2 1 48 -157 362 2 1 2 0 -157 362 -881
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
= (23 × 626391 × 44 +
543 × 53741 × 617 +
432 × 43371 × 362) mod 38209851
= 11788212 (mod 38209851)
So our answer is 11788212 (mod 38209851).
Verification
So we found that x ≡ 11788212
If this is correct, then the following statements (i.e. the original equations) are true:
576x (mod 61) ≡ 560 (mod 61)
449x (mod 711) ≡ 645 (mod 711)
797x (mod 881) ≡ 714 (mod 881)
Let's see whether that's indeed the case if we use x ≡ 11788212.