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 |
|---|---|---|---|---|---|---|
| 443 | 981 | 0 | 443 | 0 | 1 | 0 |
| 981 | 443 | 2 | 95 | 1 | 0 | 1 |
| 443 | 95 | 4 | 63 | 0 | 1 | -4 |
| 95 | 63 | 1 | 32 | 1 | -4 | 5 |
| 63 | 32 | 1 | 31 | -4 | 5 | -9 |
| 32 | 31 | 1 | 1 | 5 | -9 | 14 |
| 31 | 1 | 31 | 0 | -9 | 14 | -443 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 764 | 467 | 1 | 297 | 0 | 1 | -1 |
| 467 | 297 | 1 | 170 | 1 | -1 | 2 |
| 297 | 170 | 1 | 127 | -1 | 2 | -3 |
| 170 | 127 | 1 | 43 | 2 | -3 | 5 |
| 127 | 43 | 2 | 41 | -3 | 5 | -13 |
| 43 | 41 | 1 | 2 | 5 | -13 | 18 |
| 41 | 2 | 20 | 1 | -13 | 18 | -373 |
| 2 | 1 | 2 | 0 | 18 | -373 | 764 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 953 | 673 | 1 | 280 | 0 | 1 | -1 |
| 673 | 280 | 2 | 113 | 1 | -1 | 3 |
| 280 | 113 | 2 | 54 | -1 | 3 | -7 |
| 113 | 54 | 2 | 5 | 3 | -7 | 17 |
| 54 | 5 | 10 | 4 | -7 | 17 | -177 |
| 5 | 4 | 1 | 1 | 17 | -177 | 194 |
| 4 | 1 | 4 | 0 | -177 | 194 | -953 |
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 ≡ 824 × 981-1 (mod 443) ≡ 824 × 14 (mod 443) ≡ 18 (mod 443)x ≡ 60 × 467-1 (mod 764) ≡ 60 × 391 (mod 764) ≡ 540 (mod 764)x ≡ 264 × 673-1 (mod 953) ≡ 264 × 194 (mod 953) ≡ 707 (mod 953)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 18 (mod 443)
x ≡ 540 (mod 764)
x ≡ 707 (mod 953)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 443 × 764 × 953 = 322544756 - We calculate the numbers M1 to M3
M1=M/m1=322544756/443=728092,M2=M/m2=322544756/764=422179,M3=M/m3=322544756/953=338452 - 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 -206 mod 443 ≡ 237n b q r t1 t2 t3 443 728092 0 443 0 1 0 728092 443 1643 243 1 0 1 443 243 1 200 0 1 -1 243 200 1 43 1 -1 2 200 43 4 28 -1 2 -9 43 28 1 15 2 -9 11 28 15 1 13 -9 11 -20 15 13 1 2 11 -20 31 13 2 6 1 -20 31 -206 2 1 2 0 31 -206 443
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 227 mod 764 ≡ 227n b q r t1 t2 t3 764 422179 0 764 0 1 0 422179 764 552 451 1 0 1 764 451 1 313 0 1 -1 451 313 1 138 1 -1 2 313 138 2 37 -1 2 -5 138 37 3 27 2 -5 17 37 27 1 10 -5 17 -22 27 10 2 7 17 -22 61 10 7 1 3 -22 61 -83 7 3 2 1 61 -83 227 3 1 3 0 -83 227 -764
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 160 mod 953 ≡ 160n b q r t1 t2 t3 953 338452 0 953 0 1 0 338452 953 355 137 1 0 1 953 137 6 131 0 1 -6 137 131 1 6 1 -6 7 131 6 21 5 -6 7 -153 6 5 1 1 7 -153 160 5 1 5 0 -153 160 -953
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
= (18 × 728092 × 237 +
540 × 422179 × 227 +
707 × 338452 × 160) mod 322544756
= 249542804 (mod 322544756)
So our answer is 249542804 (mod 322544756).
Verification
So we found that x ≡ 249542804
If this is correct, then the following statements (i.e. the original equations) are true:
981x (mod 443) ≡ 824 (mod 443)
467x (mod 764) ≡ 60 (mod 764)
673x (mod 953) ≡ 264 (mod 953)
Let's see whether that's indeed the case if we use x ≡ 249542804.