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 |
|---|---|---|---|---|---|---|
| 805 | 183 | 4 | 73 | 0 | 1 | -4 |
| 183 | 73 | 2 | 37 | 1 | -4 | 9 |
| 73 | 37 | 1 | 36 | -4 | 9 | -13 |
| 37 | 36 | 1 | 1 | 9 | -13 | 22 |
| 36 | 1 | 36 | 0 | -13 | 22 | -805 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 521 | 37 | 14 | 3 | 0 | 1 | -14 |
| 37 | 3 | 12 | 1 | 1 | -14 | 169 |
| 3 | 1 | 3 | 0 | -14 | 169 | -521 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 607 | 382 | 1 | 225 | 0 | 1 | -1 |
| 382 | 225 | 1 | 157 | 1 | -1 | 2 |
| 225 | 157 | 1 | 68 | -1 | 2 | -3 |
| 157 | 68 | 2 | 21 | 2 | -3 | 8 |
| 68 | 21 | 3 | 5 | -3 | 8 | -27 |
| 21 | 5 | 4 | 1 | 8 | -27 | 116 |
| 5 | 1 | 5 | 0 | -27 | 116 | -607 |
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 ≡ 478 × 183-1 (mod 805) ≡ 478 × 22 (mod 805) ≡ 51 (mod 805)x ≡ 643 × 37-1 (mod 521) ≡ 643 × 169 (mod 521) ≡ 299 (mod 521)x ≡ 255 × 382-1 (mod 607) ≡ 255 × 116 (mod 607) ≡ 444 (mod 607)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 51 (mod 805)
x ≡ 299 (mod 521)
x ≡ 444 (mod 607)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 805 × 521 × 607 = 254578835 - We calculate the numbers M1 to M3
M1=M/m1=254578835/805=316247,M2=M/m2=254578835/521=488635,M3=M/m3=254578835/607=419405 - 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 -307 mod 805 ≡ 498n b q r t1 t2 t3 805 316247 0 805 0 1 0 316247 805 392 687 1 0 1 805 687 1 118 0 1 -1 687 118 5 97 1 -1 6 118 97 1 21 -1 6 -7 97 21 4 13 6 -7 34 21 13 1 8 -7 34 -41 13 8 1 5 34 -41 75 8 5 1 3 -41 75 -116 5 3 1 2 75 -116 191 3 2 1 1 -116 191 -307 2 1 2 0 191 -307 805
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 215 mod 521 ≡ 215n b q r t1 t2 t3 521 488635 0 521 0 1 0 488635 521 937 458 1 0 1 521 458 1 63 0 1 -1 458 63 7 17 1 -1 8 63 17 3 12 -1 8 -25 17 12 1 5 8 -25 33 12 5 2 2 -25 33 -91 5 2 2 1 33 -91 215 2 1 2 0 -91 215 -521
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -19 mod 607 ≡ 588n b q r t1 t2 t3 607 419405 0 607 0 1 0 419405 607 690 575 1 0 1 607 575 1 32 0 1 -1 575 32 17 31 1 -1 18 32 31 1 1 -1 18 -19 31 1 31 0 18 -19 607
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
= (51 × 316247 × 498 +
299 × 488635 × 215 +
444 × 419405 × 588) mod 254578835
= 10225966 (mod 254578835)
So our answer is 10225966 (mod 254578835).
Verification
So we found that x ≡ 10225966
If this is correct, then the following statements (i.e. the original equations) are true:
183x (mod 805) ≡ 478 (mod 805)
37x (mod 521) ≡ 643 (mod 521)
382x (mod 607) ≡ 255 (mod 607)
Let's see whether that's indeed the case if we use x ≡ 10225966.