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 |
|---|---|---|---|---|---|---|
| 994 | 675 | 1 | 319 | 0 | 1 | -1 |
| 675 | 319 | 2 | 37 | 1 | -1 | 3 |
| 319 | 37 | 8 | 23 | -1 | 3 | -25 |
| 37 | 23 | 1 | 14 | 3 | -25 | 28 |
| 23 | 14 | 1 | 9 | -25 | 28 | -53 |
| 14 | 9 | 1 | 5 | 28 | -53 | 81 |
| 9 | 5 | 1 | 4 | -53 | 81 | -134 |
| 5 | 4 | 1 | 1 | 81 | -134 | 215 |
| 4 | 1 | 4 | 0 | -134 | 215 | -994 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 443 | 282 | 1 | 161 | 0 | 1 | -1 |
| 282 | 161 | 1 | 121 | 1 | -1 | 2 |
| 161 | 121 | 1 | 40 | -1 | 2 | -3 |
| 121 | 40 | 3 | 1 | 2 | -3 | 11 |
| 40 | 1 | 40 | 0 | -3 | 11 | -443 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 461 | 256 | 1 | 205 | 0 | 1 | -1 |
| 256 | 205 | 1 | 51 | 1 | -1 | 2 |
| 205 | 51 | 4 | 1 | -1 | 2 | -9 |
| 51 | 1 | 51 | 0 | 2 | -9 | 461 |
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 ≡ 148 × 675-1 (mod 994) ≡ 148 × 215 (mod 994) ≡ 12 (mod 994)x ≡ 854 × 282-1 (mod 443) ≡ 854 × 11 (mod 443) ≡ 91 (mod 443)x ≡ 293 × 256-1 (mod 461) ≡ 293 × 452 (mod 461) ≡ 129 (mod 461)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 12 (mod 994)
x ≡ 91 (mod 443)
x ≡ 129 (mod 461)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 994 × 443 × 461 = 202997662 - We calculate the numbers M1 to M3
M1=M/m1=202997662/994=204223,M2=M/m2=202997662/443=458234,M3=M/m3=202997662/461=440342 - 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 -305 mod 994 ≡ 689n b q r t1 t2 t3 994 204223 0 994 0 1 0 204223 994 205 453 1 0 1 994 453 2 88 0 1 -2 453 88 5 13 1 -2 11 88 13 6 10 -2 11 -68 13 10 1 3 11 -68 79 10 3 3 1 -68 79 -305 3 1 3 0 79 -305 994
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 85 mod 443 ≡ 85n b q r t1 t2 t3 443 458234 0 443 0 1 0 458234 443 1034 172 1 0 1 443 172 2 99 0 1 -2 172 99 1 73 1 -2 3 99 73 1 26 -2 3 -5 73 26 2 21 3 -5 13 26 21 1 5 -5 13 -18 21 5 4 1 13 -18 85 5 1 5 0 -18 85 -443
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 53 mod 461 ≡ 53n b q r t1 t2 t3 461 440342 0 461 0 1 0 440342 461 955 87 1 0 1 461 87 5 26 0 1 -5 87 26 3 9 1 -5 16 26 9 2 8 -5 16 -37 9 8 1 1 16 -37 53 8 1 8 0 -37 53 -461
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
= (12 × 204223 × 689 +
91 × 458234 × 85 +
129 × 440342 × 53) mod 202997662
= 123667528 (mod 202997662)
So our answer is 123667528 (mod 202997662).
Verification
So we found that x ≡ 123667528
If this is correct, then the following statements (i.e. the original equations) are true:
675x (mod 994) ≡ 148 (mod 994)
282x (mod 443) ≡ 854 (mod 443)
256x (mod 461) ≡ 293 (mod 461)
Let's see whether that's indeed the case if we use x ≡ 123667528.