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 |
|---|---|---|---|---|---|---|
| 883 | 546 | 1 | 337 | 0 | 1 | -1 |
| 546 | 337 | 1 | 209 | 1 | -1 | 2 |
| 337 | 209 | 1 | 128 | -1 | 2 | -3 |
| 209 | 128 | 1 | 81 | 2 | -3 | 5 |
| 128 | 81 | 1 | 47 | -3 | 5 | -8 |
| 81 | 47 | 1 | 34 | 5 | -8 | 13 |
| 47 | 34 | 1 | 13 | -8 | 13 | -21 |
| 34 | 13 | 2 | 8 | 13 | -21 | 55 |
| 13 | 8 | 1 | 5 | -21 | 55 | -76 |
| 8 | 5 | 1 | 3 | 55 | -76 | 131 |
| 5 | 3 | 1 | 2 | -76 | 131 | -207 |
| 3 | 2 | 1 | 1 | 131 | -207 | 338 |
| 2 | 1 | 2 | 0 | -207 | 338 | -883 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 853 | 833 | 1 | 20 | 0 | 1 | -1 |
| 833 | 20 | 41 | 13 | 1 | -1 | 42 |
| 20 | 13 | 1 | 7 | -1 | 42 | -43 |
| 13 | 7 | 1 | 6 | 42 | -43 | 85 |
| 7 | 6 | 1 | 1 | -43 | 85 | -128 |
| 6 | 1 | 6 | 0 | 85 | -128 | 853 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 445 | 709 | 0 | 445 | 0 | 1 | 0 |
| 709 | 445 | 1 | 264 | 1 | 0 | 1 |
| 445 | 264 | 1 | 181 | 0 | 1 | -1 |
| 264 | 181 | 1 | 83 | 1 | -1 | 2 |
| 181 | 83 | 2 | 15 | -1 | 2 | -5 |
| 83 | 15 | 5 | 8 | 2 | -5 | 27 |
| 15 | 8 | 1 | 7 | -5 | 27 | -32 |
| 8 | 7 | 1 | 1 | 27 | -32 | 59 |
| 7 | 1 | 7 | 0 | -32 | 59 | -445 |
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 ≡ 897 × 546-1 (mod 883) ≡ 897 × 338 (mod 883) ≡ 317 (mod 883)x ≡ 902 × 833-1 (mod 853) ≡ 902 × 725 (mod 853) ≡ 552 (mod 853)x ≡ 694 × 709-1 (mod 445) ≡ 694 × 59 (mod 445) ≡ 6 (mod 445)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 317 (mod 883)
x ≡ 552 (mod 853)
x ≡ 6 (mod 445)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 883 × 853 × 445 = 335173555 - We calculate the numbers M1 to M3
M1=M/m1=335173555/883=379585,M2=M/m2=335173555/853=392935,M3=M/m3=335173555/445=753199 - 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 185 mod 883 ≡ 185n b q r t1 t2 t3 883 379585 0 883 0 1 0 379585 883 429 778 1 0 1 883 778 1 105 0 1 -1 778 105 7 43 1 -1 8 105 43 2 19 -1 8 -17 43 19 2 5 8 -17 42 19 5 3 4 -17 42 -143 5 4 1 1 42 -143 185 4 1 4 0 -143 185 -883
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 312 mod 853 ≡ 312n b q r t1 t2 t3 853 392935 0 853 0 1 0 392935 853 460 555 1 0 1 853 555 1 298 0 1 -1 555 298 1 257 1 -1 2 298 257 1 41 -1 2 -3 257 41 6 11 2 -3 20 41 11 3 8 -3 20 -63 11 8 1 3 20 -63 83 8 3 2 2 -63 83 -229 3 2 1 1 83 -229 312 2 1 2 0 -229 312 -853
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 189 mod 445 ≡ 189n b q r t1 t2 t3 445 753199 0 445 0 1 0 753199 445 1692 259 1 0 1 445 259 1 186 0 1 -1 259 186 1 73 1 -1 2 186 73 2 40 -1 2 -5 73 40 1 33 2 -5 7 40 33 1 7 -5 7 -12 33 7 4 5 7 -12 55 7 5 1 2 -12 55 -67 5 2 2 1 55 -67 189 2 1 2 0 -67 189 -445
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
= (317 × 379585 × 185 +
552 × 392935 × 312 +
6 × 753199 × 189) mod 335173555
= 290867581 (mod 335173555)
So our answer is 290867581 (mod 335173555).
Verification
So we found that x ≡ 290867581
If this is correct, then the following statements (i.e. the original equations) are true:
546x (mod 883) ≡ 897 (mod 883)
833x (mod 853) ≡ 902 (mod 853)
709x (mod 445) ≡ 694 (mod 445)
Let's see whether that's indeed the case if we use x ≡ 290867581.