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" .
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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 |
|---|---|---|---|---|---|---|
| 293 | 928 | 0 | 293 | 0 | 1 | 0 |
| 928 | 293 | 3 | 49 | 1 | 0 | 1 |
| 293 | 49 | 5 | 48 | 0 | 1 | -5 |
| 49 | 48 | 1 | 1 | 1 | -5 | 6 |
| 48 | 1 | 48 | 0 | -5 | 6 | -293 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 791 | 549 | 1 | 242 | 0 | 1 | -1 |
| 549 | 242 | 2 | 65 | 1 | -1 | 3 |
| 242 | 65 | 3 | 47 | -1 | 3 | -10 |
| 65 | 47 | 1 | 18 | 3 | -10 | 13 |
| 47 | 18 | 2 | 11 | -10 | 13 | -36 |
| 18 | 11 | 1 | 7 | 13 | -36 | 49 |
| 11 | 7 | 1 | 4 | -36 | 49 | -85 |
| 7 | 4 | 1 | 3 | 49 | -85 | 134 |
| 4 | 3 | 1 | 1 | -85 | 134 | -219 |
| 3 | 1 | 3 | 0 | 134 | -219 | 791 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 499 | 381 | 1 | 118 | 0 | 1 | -1 |
| 381 | 118 | 3 | 27 | 1 | -1 | 4 |
| 118 | 27 | 4 | 10 | -1 | 4 | -17 |
| 27 | 10 | 2 | 7 | 4 | -17 | 38 |
| 10 | 7 | 1 | 3 | -17 | 38 | -55 |
| 7 | 3 | 2 | 1 | 38 | -55 | 148 |
| 3 | 1 | 3 | 0 | -55 | 148 | -499 |
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 ≡ 460 × 928-1 (mod 293) ≡ 460 × 6 (mod 293) ≡ 123 (mod 293)x ≡ 165 × 549-1 (mod 791) ≡ 165 × 572 (mod 791) ≡ 251 (mod 791)x ≡ 165 × 381-1 (mod 499) ≡ 165 × 148 (mod 499) ≡ 468 (mod 499)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 123 (mod 293)
x ≡ 251 (mod 791)
x ≡ 468 (mod 499)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 293 × 791 × 499 = 115649737 - We calculate the numbers M1 to M3
M1=M/m1=115649737/293=394709,M2=M/m2=115649737/791=146207,M3=M/m3=115649737/499=231763 - 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 54 mod 293 ≡ 54n b q r t1 t2 t3 293 394709 0 293 0 1 0 394709 293 1347 38 1 0 1 293 38 7 27 0 1 -7 38 27 1 11 1 -7 8 27 11 2 5 -7 8 -23 11 5 2 1 8 -23 54 5 1 5 0 -23 54 -293
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 241 mod 791 ≡ 241n b q r t1 t2 t3 791 146207 0 791 0 1 0 146207 791 184 663 1 0 1 791 663 1 128 0 1 -1 663 128 5 23 1 -1 6 128 23 5 13 -1 6 -31 23 13 1 10 6 -31 37 13 10 1 3 -31 37 -68 10 3 3 1 37 -68 241 3 1 3 0 -68 241 -791
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -244 mod 499 ≡ 255n b q r t1 t2 t3 499 231763 0 499 0 1 0 231763 499 464 227 1 0 1 499 227 2 45 0 1 -2 227 45 5 2 1 -2 11 45 2 22 1 -2 11 -244 2 1 2 0 11 -244 499
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
= (123 × 394709 × 54 +
251 × 146207 × 241 +
468 × 231763 × 255) mod 115649737
= 34850129 (mod 115649737)
So our answer is 34850129 (mod 115649737).
Verification
So we found that x ≡ 34850129
If this is correct, then the following statements (i.e. the original equations) are true:
928x (mod 293) ≡ 460 (mod 293)
549x (mod 791) ≡ 165 (mod 791)
381x (mod 499) ≡ 165 (mod 499)
Let's see whether that's indeed the case if we use x ≡ 34850129.