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 |
|---|---|---|---|---|---|---|
| 529 | 344 | 1 | 185 | 0 | 1 | -1 |
| 344 | 185 | 1 | 159 | 1 | -1 | 2 |
| 185 | 159 | 1 | 26 | -1 | 2 | -3 |
| 159 | 26 | 6 | 3 | 2 | -3 | 20 |
| 26 | 3 | 8 | 2 | -3 | 20 | -163 |
| 3 | 2 | 1 | 1 | 20 | -163 | 183 |
| 2 | 1 | 2 | 0 | -163 | 183 | -529 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 974 | 573 | 1 | 401 | 0 | 1 | -1 |
| 573 | 401 | 1 | 172 | 1 | -1 | 2 |
| 401 | 172 | 2 | 57 | -1 | 2 | -5 |
| 172 | 57 | 3 | 1 | 2 | -5 | 17 |
| 57 | 1 | 57 | 0 | -5 | 17 | -974 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 383 | 54 | 7 | 5 | 0 | 1 | -7 |
| 54 | 5 | 10 | 4 | 1 | -7 | 71 |
| 5 | 4 | 1 | 1 | -7 | 71 | -78 |
| 4 | 1 | 4 | 0 | 71 | -78 | 383 |
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 ≡ 252 × 344-1 (mod 529) ≡ 252 × 183 (mod 529) ≡ 93 (mod 529)x ≡ 368 × 573-1 (mod 974) ≡ 368 × 17 (mod 974) ≡ 412 (mod 974)x ≡ 970 × 54-1 (mod 383) ≡ 970 × 305 (mod 383) ≡ 174 (mod 383)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 93 (mod 529)
x ≡ 412 (mod 974)
x ≡ 174 (mod 383)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 529 × 974 × 383 = 197339218 - We calculate the numbers M1 to M3
M1=M/m1=197339218/529=373042,M2=M/m2=197339218/974=202607,M3=M/m3=197339218/383=515246 - 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 60 mod 529 ≡ 60n b q r t1 t2 t3 529 373042 0 529 0 1 0 373042 529 705 97 1 0 1 529 97 5 44 0 1 -5 97 44 2 9 1 -5 11 44 9 4 8 -5 11 -49 9 8 1 1 11 -49 60 8 1 8 0 -49 60 -529
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 65 mod 974 ≡ 65n b q r t1 t2 t3 974 202607 0 974 0 1 0 202607 974 208 15 1 0 1 974 15 64 14 0 1 -64 15 14 1 1 1 -64 65 14 1 14 0 -64 65 -974
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -69 mod 383 ≡ 314n b q r t1 t2 t3 383 515246 0 383 0 1 0 515246 383 1345 111 1 0 1 383 111 3 50 0 1 -3 111 50 2 11 1 -3 7 50 11 4 6 -3 7 -31 11 6 1 5 7 -31 38 6 5 1 1 -31 38 -69 5 1 5 0 38 -69 383
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
= (93 × 373042 × 60 +
412 × 202607 × 65 +
174 × 515246 × 314) mod 197339218
= 137311036 (mod 197339218)
So our answer is 137311036 (mod 197339218).
Verification
So we found that x ≡ 137311036
If this is correct, then the following statements (i.e. the original equations) are true:
344x (mod 529) ≡ 252 (mod 529)
573x (mod 974) ≡ 368 (mod 974)
54x (mod 383) ≡ 970 (mod 383)
Let's see whether that's indeed the case if we use x ≡ 137311036.