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!
Do you have too many rows? Use one of these buttons to remove a row. You can always add a row again using the yellow "Add a row" button.
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 |
|---|---|---|---|---|---|---|
| 13 | 851 | 0 | 13 | 0 | 1 | 0 |
| 851 | 13 | 65 | 6 | 1 | 0 | 1 |
| 13 | 6 | 2 | 1 | 0 | 1 | -2 |
| 6 | 1 | 6 | 0 | 1 | -2 | 13 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 421 | 429 | 0 | 421 | 0 | 1 | 0 |
| 429 | 421 | 1 | 8 | 1 | 0 | 1 |
| 421 | 8 | 52 | 5 | 0 | 1 | -52 |
| 8 | 5 | 1 | 3 | 1 | -52 | 53 |
| 5 | 3 | 1 | 2 | -52 | 53 | -105 |
| 3 | 2 | 1 | 1 | 53 | -105 | 158 |
| 2 | 1 | 2 | 0 | -105 | 158 | -421 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 857 | 657 | 1 | 200 | 0 | 1 | -1 |
| 657 | 200 | 3 | 57 | 1 | -1 | 4 |
| 200 | 57 | 3 | 29 | -1 | 4 | -13 |
| 57 | 29 | 1 | 28 | 4 | -13 | 17 |
| 29 | 28 | 1 | 1 | -13 | 17 | -30 |
| 28 | 1 | 28 | 0 | 17 | -30 | 857 |
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 ≡ 370 × 851-1 (mod 13) ≡ 370 × 11 (mod 13) ≡ 1 (mod 13)x ≡ 398 × 429-1 (mod 421) ≡ 398 × 158 (mod 421) ≡ 155 (mod 421)x ≡ 483 × 657-1 (mod 857) ≡ 483 × 827 (mod 857) ≡ 79 (mod 857)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 1 (mod 13)
x ≡ 155 (mod 421)
x ≡ 79 (mod 857)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 13 × 421 × 857 = 4690361 - We calculate the numbers M1 to M3
M1=M/m1=4690361/13=360797,M2=M/m2=4690361/421=11141,M3=M/m3=4690361/857=5473 - 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 5 mod 13 ≡ 5n b q r t1 t2 t3 13 360797 0 13 0 1 0 360797 13 27753 8 1 0 1 13 8 1 5 0 1 -1 8 5 1 3 1 -1 2 5 3 1 2 -1 2 -3 3 2 1 1 2 -3 5 2 1 2 0 -3 5 -13
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 95 mod 421 ≡ 95n b q r t1 t2 t3 421 11141 0 421 0 1 0 11141 421 26 195 1 0 1 421 195 2 31 0 1 -2 195 31 6 9 1 -2 13 31 9 3 4 -2 13 -41 9 4 2 1 13 -41 95 4 1 4 0 -41 95 -421
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 334 mod 857 ≡ 334n b q r t1 t2 t3 857 5473 0 857 0 1 0 5473 857 6 331 1 0 1 857 331 2 195 0 1 -2 331 195 1 136 1 -2 3 195 136 1 59 -2 3 -5 136 59 2 18 3 -5 13 59 18 3 5 -5 13 -44 18 5 3 3 13 -44 145 5 3 1 2 -44 145 -189 3 2 1 1 145 -189 334 2 1 2 0 -189 334 -857
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
= (1 × 360797 × 5 +
155 × 11141 × 95 +
79 × 5473 × 334) mod 4690361
= 701962 (mod 4690361)
So our answer is 701962 (mod 4690361).
Verification
So we found that x ≡ 701962
If this is correct, then the following statements (i.e. the original equations) are true:
851x (mod 13) ≡ 370 (mod 13)
429x (mod 421) ≡ 398 (mod 421)
657x (mod 857) ≡ 483 (mod 857)
Let's see whether that's indeed the case if we use x ≡ 701962.