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 |
|---|---|---|---|---|---|---|
| 961 | 436 | 2 | 89 | 0 | 1 | -2 |
| 436 | 89 | 4 | 80 | 1 | -2 | 9 |
| 89 | 80 | 1 | 9 | -2 | 9 | -11 |
| 80 | 9 | 8 | 8 | 9 | -11 | 97 |
| 9 | 8 | 1 | 1 | -11 | 97 | -108 |
| 8 | 1 | 8 | 0 | 97 | -108 | 961 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 407 | 619 | 0 | 407 | 0 | 1 | 0 |
| 619 | 407 | 1 | 212 | 1 | 0 | 1 |
| 407 | 212 | 1 | 195 | 0 | 1 | -1 |
| 212 | 195 | 1 | 17 | 1 | -1 | 2 |
| 195 | 17 | 11 | 8 | -1 | 2 | -23 |
| 17 | 8 | 2 | 1 | 2 | -23 | 48 |
| 8 | 1 | 8 | 0 | -23 | 48 | -407 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 611 | 668 | 0 | 611 | 0 | 1 | 0 |
| 668 | 611 | 1 | 57 | 1 | 0 | 1 |
| 611 | 57 | 10 | 41 | 0 | 1 | -10 |
| 57 | 41 | 1 | 16 | 1 | -10 | 11 |
| 41 | 16 | 2 | 9 | -10 | 11 | -32 |
| 16 | 9 | 1 | 7 | 11 | -32 | 43 |
| 9 | 7 | 1 | 2 | -32 | 43 | -75 |
| 7 | 2 | 3 | 1 | 43 | -75 | 268 |
| 2 | 1 | 2 | 0 | -75 | 268 | -611 |
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 ≡ 604 × 436-1 (mod 961) ≡ 604 × 853 (mod 961) ≡ 116 (mod 961)x ≡ 551 × 619-1 (mod 407) ≡ 551 × 48 (mod 407) ≡ 400 (mod 407)x ≡ 433 × 668-1 (mod 611) ≡ 433 × 268 (mod 611) ≡ 565 (mod 611)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 116 (mod 961)
x ≡ 400 (mod 407)
x ≡ 565 (mod 611)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 961 × 407 × 611 = 238978597 - We calculate the numbers M1 to M3
M1=M/m1=238978597/961=248677,M2=M/m2=238978597/407=587171,M3=M/m3=238978597/611=391127 - 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 316 mod 961 ≡ 316n b q r t1 t2 t3 961 248677 0 961 0 1 0 248677 961 258 739 1 0 1 961 739 1 222 0 1 -1 739 222 3 73 1 -1 4 222 73 3 3 -1 4 -13 73 3 24 1 4 -13 316 3 1 3 0 -13 316 -961
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 72 mod 407 ≡ 72n b q r t1 t2 t3 407 587171 0 407 0 1 0 587171 407 1442 277 1 0 1 407 277 1 130 0 1 -1 277 130 2 17 1 -1 3 130 17 7 11 -1 3 -22 17 11 1 6 3 -22 25 11 6 1 5 -22 25 -47 6 5 1 1 25 -47 72 5 1 5 0 -47 72 -407
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 302 mod 611 ≡ 302n b q r t1 t2 t3 611 391127 0 611 0 1 0 391127 611 640 87 1 0 1 611 87 7 2 0 1 -7 87 2 43 1 1 -7 302 2 1 2 0 -7 302 -611
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
= (116 × 248677 × 316 +
400 × 587171 × 72 +
565 × 391127 × 302) mod 238978597
= 40333286 (mod 238978597)
So our answer is 40333286 (mod 238978597).
Verification
So we found that x ≡ 40333286
If this is correct, then the following statements (i.e. the original equations) are true:
436x (mod 961) ≡ 604 (mod 961)
619x (mod 407) ≡ 551 (mod 407)
668x (mod 611) ≡ 433 (mod 611)
Let's see whether that's indeed the case if we use x ≡ 40333286.