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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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 |
|---|---|---|---|---|---|---|
| 533 | 165 | 3 | 38 | 0 | 1 | -3 |
| 165 | 38 | 4 | 13 | 1 | -3 | 13 |
| 38 | 13 | 2 | 12 | -3 | 13 | -29 |
| 13 | 12 | 1 | 1 | 13 | -29 | 42 |
| 12 | 1 | 12 | 0 | -29 | 42 | -533 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 331 | 455 | 0 | 331 | 0 | 1 | 0 |
| 455 | 331 | 1 | 124 | 1 | 0 | 1 |
| 331 | 124 | 2 | 83 | 0 | 1 | -2 |
| 124 | 83 | 1 | 41 | 1 | -2 | 3 |
| 83 | 41 | 2 | 1 | -2 | 3 | -8 |
| 41 | 1 | 41 | 0 | 3 | -8 | 331 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 737 | 41 | 17 | 40 | 0 | 1 | -17 |
| 41 | 40 | 1 | 1 | 1 | -17 | 18 |
| 40 | 1 | 40 | 0 | -17 | 18 | -737 |
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 ≡ 969 × 165-1 (mod 533) ≡ 969 × 42 (mod 533) ≡ 190 (mod 533)x ≡ 211 × 455-1 (mod 331) ≡ 211 × 323 (mod 331) ≡ 298 (mod 331)x ≡ 303 × 41-1 (mod 737) ≡ 303 × 18 (mod 737) ≡ 295 (mod 737)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 190 (mod 533)
x ≡ 298 (mod 331)
x ≡ 295 (mod 737)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 533 × 331 × 737 = 130023751 - We calculate the numbers M1 to M3
M1=M/m1=130023751/533=243947,M2=M/m2=130023751/331=392821,M3=M/m3=130023751/737=176423 - 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 150 mod 533 ≡ 150n b q r t1 t2 t3 533 243947 0 533 0 1 0 243947 533 457 366 1 0 1 533 366 1 167 0 1 -1 366 167 2 32 1 -1 3 167 32 5 7 -1 3 -16 32 7 4 4 3 -16 67 7 4 1 3 -16 67 -83 4 3 1 1 67 -83 150 3 1 3 0 -83 150 -533
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 135 mod 331 ≡ 135n b q r t1 t2 t3 331 392821 0 331 0 1 0 392821 331 1186 255 1 0 1 331 255 1 76 0 1 -1 255 76 3 27 1 -1 4 76 27 2 22 -1 4 -9 27 22 1 5 4 -9 13 22 5 4 2 -9 13 -61 5 2 2 1 13 -61 135 2 1 2 0 -61 135 -331
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 229 mod 737 ≡ 229n b q r t1 t2 t3 737 176423 0 737 0 1 0 176423 737 239 280 1 0 1 737 280 2 177 0 1 -2 280 177 1 103 1 -2 3 177 103 1 74 -2 3 -5 103 74 1 29 3 -5 8 74 29 2 16 -5 8 -21 29 16 1 13 8 -21 29 16 13 1 3 -21 29 -50 13 3 4 1 29 -50 229 3 1 3 0 -50 229 -737
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
= (190 × 243947 × 150 +
298 × 392821 × 135 +
295 × 176423 × 229) mod 130023751
= 87616329 (mod 130023751)
So our answer is 87616329 (mod 130023751).
Verification
So we found that x ≡ 87616329
If this is correct, then the following statements (i.e. the original equations) are true:
165x (mod 533) ≡ 969 (mod 533)
455x (mod 331) ≡ 211 (mod 331)
41x (mod 737) ≡ 303 (mod 737)
Let's see whether that's indeed the case if we use x ≡ 87616329.