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.
This fills all textboxes with random numbers. If you fill in random numbers yourself, it is very likely that those numbers do not have a solution. To avoid disappointment, use this button instead! It only uses random numbers that do have a solution.
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 |
|---|---|---|---|---|---|---|
| 463 | 954 | 0 | 463 | 0 | 1 | 0 |
| 954 | 463 | 2 | 28 | 1 | 0 | 1 |
| 463 | 28 | 16 | 15 | 0 | 1 | -16 |
| 28 | 15 | 1 | 13 | 1 | -16 | 17 |
| 15 | 13 | 1 | 2 | -16 | 17 | -33 |
| 13 | 2 | 6 | 1 | 17 | -33 | 215 |
| 2 | 1 | 2 | 0 | -33 | 215 | -463 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 543 | 544 | 0 | 543 | 0 | 1 | 0 |
| 544 | 543 | 1 | 1 | 1 | 0 | 1 |
| 543 | 1 | 543 | 0 | 0 | 1 | -543 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 821 | 926 | 0 | 821 | 0 | 1 | 0 |
| 926 | 821 | 1 | 105 | 1 | 0 | 1 |
| 821 | 105 | 7 | 86 | 0 | 1 | -7 |
| 105 | 86 | 1 | 19 | 1 | -7 | 8 |
| 86 | 19 | 4 | 10 | -7 | 8 | -39 |
| 19 | 10 | 1 | 9 | 8 | -39 | 47 |
| 10 | 9 | 1 | 1 | -39 | 47 | -86 |
| 9 | 1 | 9 | 0 | 47 | -86 | 821 |
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 ≡ 971 × 954-1 (mod 463) ≡ 971 × 215 (mod 463) ≡ 415 (mod 463)x ≡ 916 × 544-1 (mod 543) ≡ 916 × 1 (mod 543) ≡ 373 (mod 543)x ≡ 873 × 926-1 (mod 821) ≡ 873 × 735 (mod 821) ≡ 454 (mod 821)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 415 (mod 463)
x ≡ 373 (mod 543)
x ≡ 454 (mod 821)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 463 × 543 × 821 = 206406789 - We calculate the numbers M1 to M3
M1=M/m1=206406789/463=445803,M2=M/m2=206406789/543=380123,M3=M/m3=206406789/821=251409 - 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 7 mod 463 ≡ 7n b q r t1 t2 t3 463 445803 0 463 0 1 0 445803 463 962 397 1 0 1 463 397 1 66 0 1 -1 397 66 6 1 1 -1 7 66 1 66 0 -1 7 -463
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -118 mod 543 ≡ 425n b q r t1 t2 t3 543 380123 0 543 0 1 0 380123 543 700 23 1 0 1 543 23 23 14 0 1 -23 23 14 1 9 1 -23 24 14 9 1 5 -23 24 -47 9 5 1 4 24 -47 71 5 4 1 1 -47 71 -118 4 1 4 0 71 -118 543
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 166 mod 821 ≡ 166n b q r t1 t2 t3 821 251409 0 821 0 1 0 251409 821 306 183 1 0 1 821 183 4 89 0 1 -4 183 89 2 5 1 -4 9 89 5 17 4 -4 9 -157 5 4 1 1 9 -157 166 4 1 4 0 -157 166 -821
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
= (415 × 445803 × 7 +
373 × 380123 × 425 +
454 × 251409 × 166) mod 206406789
= 2596456 (mod 206406789)
So our answer is 2596456 (mod 206406789).
Verification
So we found that x ≡ 2596456
If this is correct, then the following statements (i.e. the original equations) are true:
954x (mod 463) ≡ 971 (mod 463)
544x (mod 543) ≡ 916 (mod 543)
926x (mod 821) ≡ 873 (mod 821)
Let's see whether that's indeed the case if we use x ≡ 2596456.