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 |
|---|---|---|---|---|---|---|
| 749 | 72 | 10 | 29 | 0 | 1 | -10 |
| 72 | 29 | 2 | 14 | 1 | -10 | 21 |
| 29 | 14 | 2 | 1 | -10 | 21 | -52 |
| 14 | 1 | 14 | 0 | 21 | -52 | 749 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 545 | 586 | 0 | 545 | 0 | 1 | 0 |
| 586 | 545 | 1 | 41 | 1 | 0 | 1 |
| 545 | 41 | 13 | 12 | 0 | 1 | -13 |
| 41 | 12 | 3 | 5 | 1 | -13 | 40 |
| 12 | 5 | 2 | 2 | -13 | 40 | -93 |
| 5 | 2 | 2 | 1 | 40 | -93 | 226 |
| 2 | 1 | 2 | 0 | -93 | 226 | -545 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 319 | 31 | 10 | 9 | 0 | 1 | -10 |
| 31 | 9 | 3 | 4 | 1 | -10 | 31 |
| 9 | 4 | 2 | 1 | -10 | 31 | -72 |
| 4 | 1 | 4 | 0 | 31 | -72 | 319 |
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 ≡ 636 × 72-1 (mod 749) ≡ 636 × 697 (mod 749) ≡ 633 (mod 749)x ≡ 920 × 586-1 (mod 545) ≡ 920 × 226 (mod 545) ≡ 275 (mod 545)x ≡ 508 × 31-1 (mod 319) ≡ 508 × 247 (mod 319) ≡ 109 (mod 319)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 633 (mod 749)
x ≡ 275 (mod 545)
x ≡ 109 (mod 319)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 749 × 545 × 319 = 130217395 - We calculate the numbers M1 to M3
M1=M/m1=130217395/749=173855,M2=M/m2=130217395/545=238931,M3=M/m3=130217395/319=408205 - 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 -198 mod 749 ≡ 551n b q r t1 t2 t3 749 173855 0 749 0 1 0 173855 749 232 87 1 0 1 749 87 8 53 0 1 -8 87 53 1 34 1 -8 9 53 34 1 19 -8 9 -17 34 19 1 15 9 -17 26 19 15 1 4 -17 26 -43 15 4 3 3 26 -43 155 4 3 1 1 -43 155 -198 3 1 3 0 155 -198 749
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -254 mod 545 ≡ 291n b q r t1 t2 t3 545 238931 0 545 0 1 0 238931 545 438 221 1 0 1 545 221 2 103 0 1 -2 221 103 2 15 1 -2 5 103 15 6 13 -2 5 -32 15 13 1 2 5 -32 37 13 2 6 1 -32 37 -254 2 1 2 0 37 -254 545
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -86 mod 319 ≡ 233n b q r t1 t2 t3 319 408205 0 319 0 1 0 408205 319 1279 204 1 0 1 319 204 1 115 0 1 -1 204 115 1 89 1 -1 2 115 89 1 26 -1 2 -3 89 26 3 11 2 -3 11 26 11 2 4 -3 11 -25 11 4 2 3 11 -25 61 4 3 1 1 -25 61 -86 3 1 3 0 61 -86 319
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
= (633 × 173855 × 551 +
275 × 238931 × 291 +
109 × 408205 × 233) mod 130217395
= 14866785 (mod 130217395)
So our answer is 14866785 (mod 130217395).
Verification
So we found that x ≡ 14866785
If this is correct, then the following statements (i.e. the original equations) are true:
72x (mod 749) ≡ 636 (mod 749)
586x (mod 545) ≡ 920 (mod 545)
31x (mod 319) ≡ 508 (mod 319)
Let's see whether that's indeed the case if we use x ≡ 14866785.