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 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 |
|---|---|---|---|---|---|---|
| 461 | 637 | 0 | 461 | 0 | 1 | 0 |
| 637 | 461 | 1 | 176 | 1 | 0 | 1 |
| 461 | 176 | 2 | 109 | 0 | 1 | -2 |
| 176 | 109 | 1 | 67 | 1 | -2 | 3 |
| 109 | 67 | 1 | 42 | -2 | 3 | -5 |
| 67 | 42 | 1 | 25 | 3 | -5 | 8 |
| 42 | 25 | 1 | 17 | -5 | 8 | -13 |
| 25 | 17 | 1 | 8 | 8 | -13 | 21 |
| 17 | 8 | 2 | 1 | -13 | 21 | -55 |
| 8 | 1 | 8 | 0 | 21 | -55 | 461 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 557 | 181 | 3 | 14 | 0 | 1 | -3 |
| 181 | 14 | 12 | 13 | 1 | -3 | 37 |
| 14 | 13 | 1 | 1 | -3 | 37 | -40 |
| 13 | 1 | 13 | 0 | 37 | -40 | 557 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 997 | 813 | 1 | 184 | 0 | 1 | -1 |
| 813 | 184 | 4 | 77 | 1 | -1 | 5 |
| 184 | 77 | 2 | 30 | -1 | 5 | -11 |
| 77 | 30 | 2 | 17 | 5 | -11 | 27 |
| 30 | 17 | 1 | 13 | -11 | 27 | -38 |
| 17 | 13 | 1 | 4 | 27 | -38 | 65 |
| 13 | 4 | 3 | 1 | -38 | 65 | -233 |
| 4 | 1 | 4 | 0 | 65 | -233 | 997 |
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 ≡ 655 × 637-1 (mod 461) ≡ 655 × 406 (mod 461) ≡ 394 (mod 461)x ≡ 807 × 181-1 (mod 557) ≡ 807 × 517 (mod 557) ≡ 26 (mod 557)x ≡ 798 × 813-1 (mod 997) ≡ 798 × 764 (mod 997) ≡ 505 (mod 997)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 394 (mod 461)
x ≡ 26 (mod 557)
x ≡ 505 (mod 997)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 461 × 557 × 997 = 256006669 - We calculate the numbers M1 to M3
M1=M/m1=256006669/461=555329,M2=M/m2=256006669/557=459617,M3=M/m3=256006669/997=256777 - 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 55 mod 461 ≡ 55n b q r t1 t2 t3 461 555329 0 461 0 1 0 555329 461 1204 285 1 0 1 461 285 1 176 0 1 -1 285 176 1 109 1 -1 2 176 109 1 67 -1 2 -3 109 67 1 42 2 -3 5 67 42 1 25 -3 5 -8 42 25 1 17 5 -8 13 25 17 1 8 -8 13 -21 17 8 2 1 13 -21 55 8 1 8 0 -21 55 -461
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -224 mod 557 ≡ 333n b q r t1 t2 t3 557 459617 0 557 0 1 0 459617 557 825 92 1 0 1 557 92 6 5 0 1 -6 92 5 18 2 1 -6 109 5 2 2 1 -6 109 -224 2 1 2 0 109 -224 557
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 282 mod 997 ≡ 282n b q r t1 t2 t3 997 256777 0 997 0 1 0 256777 997 257 548 1 0 1 997 548 1 449 0 1 -1 548 449 1 99 1 -1 2 449 99 4 53 -1 2 -9 99 53 1 46 2 -9 11 53 46 1 7 -9 11 -20 46 7 6 4 11 -20 131 7 4 1 3 -20 131 -151 4 3 1 1 131 -151 282 3 1 3 0 -151 282 -997
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
= (394 × 555329 × 55 +
26 × 459617 × 333 +
505 × 256777 × 282) mod 256006669
= 99588841 (mod 256006669)
So our answer is 99588841 (mod 256006669).
Verification
So we found that x ≡ 99588841
If this is correct, then the following statements (i.e. the original equations) are true:
637x (mod 461) ≡ 655 (mod 461)
181x (mod 557) ≡ 807 (mod 557)
813x (mod 997) ≡ 798 (mod 997)
Let's see whether that's indeed the case if we use x ≡ 99588841.