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.
This fills all textboxes with random numbers. If you fill in random numbers yourself, it is very likely that those numbers do not have a solution. To avoid disappointment, use this button instead! It only uses random numbers that do have a solution.
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.
Do you want to use more equations? Go ahead and use this button. It adds another row that you can fill in. Not sure what numbers to put in this newly added row? Use the random numbers button again!
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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 |
|---|---|---|---|---|---|---|
| 431 | 778 | 0 | 431 | 0 | 1 | 0 |
| 778 | 431 | 1 | 347 | 1 | 0 | 1 |
| 431 | 347 | 1 | 84 | 0 | 1 | -1 |
| 347 | 84 | 4 | 11 | 1 | -1 | 5 |
| 84 | 11 | 7 | 7 | -1 | 5 | -36 |
| 11 | 7 | 1 | 4 | 5 | -36 | 41 |
| 7 | 4 | 1 | 3 | -36 | 41 | -77 |
| 4 | 3 | 1 | 1 | 41 | -77 | 118 |
| 3 | 1 | 3 | 0 | -77 | 118 | -431 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 421 | 481 | 0 | 421 | 0 | 1 | 0 |
| 481 | 421 | 1 | 60 | 1 | 0 | 1 |
| 421 | 60 | 7 | 1 | 0 | 1 | -7 |
| 60 | 1 | 60 | 0 | 1 | -7 | 421 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 401 | 104 | 3 | 89 | 0 | 1 | -3 |
| 104 | 89 | 1 | 15 | 1 | -3 | 4 |
| 89 | 15 | 5 | 14 | -3 | 4 | -23 |
| 15 | 14 | 1 | 1 | 4 | -23 | 27 |
| 14 | 1 | 14 | 0 | -23 | 27 | -401 |
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 ≡ 28 × 778-1 (mod 431) ≡ 28 × 118 (mod 431) ≡ 287 (mod 431)x ≡ 268 × 481-1 (mod 421) ≡ 268 × 414 (mod 421) ≡ 229 (mod 421)x ≡ 339 × 104-1 (mod 401) ≡ 339 × 27 (mod 401) ≡ 331 (mod 401)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 287 (mod 431)
x ≡ 229 (mod 421)
x ≡ 331 (mod 401)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 431 × 421 × 401 = 72761851 - We calculate the numbers M1 to M3
M1=M/m1=72761851/431=168821,M2=M/m2=72761851/421=172831,M3=M/m3=72761851/401=181451 - 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 -102 mod 431 ≡ 329n b q r t1 t2 t3 431 168821 0 431 0 1 0 168821 431 391 300 1 0 1 431 300 1 131 0 1 -1 300 131 2 38 1 -1 3 131 38 3 17 -1 3 -10 38 17 2 4 3 -10 23 17 4 4 1 -10 23 -102 4 1 4 0 23 -102 431
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -40 mod 421 ≡ 381n b q r t1 t2 t3 421 172831 0 421 0 1 0 172831 421 410 221 1 0 1 421 221 1 200 0 1 -1 221 200 1 21 1 -1 2 200 21 9 11 -1 2 -19 21 11 1 10 2 -19 21 11 10 1 1 -19 21 -40 10 1 10 0 21 -40 421
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 133 mod 401 ≡ 133n b q r t1 t2 t3 401 181451 0 401 0 1 0 181451 401 452 199 1 0 1 401 199 2 3 0 1 -2 199 3 66 1 1 -2 133 3 1 3 0 -2 133 -401
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
= (287 × 168821 × 329 +
229 × 172831 × 381 +
331 × 181451 × 133) mod 72761851
= 7582439 (mod 72761851)
So our answer is 7582439 (mod 72761851).
Verification
So we found that x ≡ 7582439
If this is correct, then the following statements (i.e. the original equations) are true:
778x (mod 431) ≡ 28 (mod 431)
481x (mod 421) ≡ 268 (mod 421)
104x (mod 401) ≡ 339 (mod 401)
Let's see whether that's indeed the case if we use x ≡ 7582439.