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 |
|---|---|---|---|---|---|---|
| 689 | 745 | 0 | 689 | 0 | 1 | 0 |
| 745 | 689 | 1 | 56 | 1 | 0 | 1 |
| 689 | 56 | 12 | 17 | 0 | 1 | -12 |
| 56 | 17 | 3 | 5 | 1 | -12 | 37 |
| 17 | 5 | 3 | 2 | -12 | 37 | -123 |
| 5 | 2 | 2 | 1 | 37 | -123 | 283 |
| 2 | 1 | 2 | 0 | -123 | 283 | -689 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 943 | 677 | 1 | 266 | 0 | 1 | -1 |
| 677 | 266 | 2 | 145 | 1 | -1 | 3 |
| 266 | 145 | 1 | 121 | -1 | 3 | -4 |
| 145 | 121 | 1 | 24 | 3 | -4 | 7 |
| 121 | 24 | 5 | 1 | -4 | 7 | -39 |
| 24 | 1 | 24 | 0 | 7 | -39 | 943 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 661 | 109 | 6 | 7 | 0 | 1 | -6 |
| 109 | 7 | 15 | 4 | 1 | -6 | 91 |
| 7 | 4 | 1 | 3 | -6 | 91 | -97 |
| 4 | 3 | 1 | 1 | 91 | -97 | 188 |
| 3 | 1 | 3 | 0 | -97 | 188 | -661 |
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 ≡ 640 × 745-1 (mod 689) ≡ 640 × 283 (mod 689) ≡ 602 (mod 689)x ≡ 365 × 677-1 (mod 943) ≡ 365 × 904 (mod 943) ≡ 853 (mod 943)x ≡ 957 × 109-1 (mod 661) ≡ 957 × 188 (mod 661) ≡ 124 (mod 661)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 602 (mod 689)
x ≡ 853 (mod 943)
x ≡ 124 (mod 661)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 689 × 943 × 661 = 429469547 - We calculate the numbers M1 to M3
M1=M/m1=429469547/689=623323,M2=M/m2=429469547/943=455429,M3=M/m3=429469547/661=649727 - 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 90 mod 689 ≡ 90n b q r t1 t2 t3 689 623323 0 689 0 1 0 623323 689 904 467 1 0 1 689 467 1 222 0 1 -1 467 222 2 23 1 -1 3 222 23 9 15 -1 3 -28 23 15 1 8 3 -28 31 15 8 1 7 -28 31 -59 8 7 1 1 31 -59 90 7 1 7 0 -59 90 -689
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 165 mod 943 ≡ 165n b q r t1 t2 t3 943 455429 0 943 0 1 0 455429 943 482 903 1 0 1 943 903 1 40 0 1 -1 903 40 22 23 1 -1 23 40 23 1 17 -1 23 -24 23 17 1 6 23 -24 47 17 6 2 5 -24 47 -118 6 5 1 1 47 -118 165 5 1 5 0 -118 165 -943
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -202 mod 661 ≡ 459n b q r t1 t2 t3 661 649727 0 661 0 1 0 649727 661 982 625 1 0 1 661 625 1 36 0 1 -1 625 36 17 13 1 -1 18 36 13 2 10 -1 18 -37 13 10 1 3 18 -37 55 10 3 3 1 -37 55 -202 3 1 3 0 55 -202 661
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
= (602 × 623323 × 90 +
853 × 455429 × 165 +
124 × 649727 × 459) mod 429469547
= 426888466 (mod 429469547)
So our answer is 426888466 (mod 429469547).
Verification
So we found that x ≡ 426888466
If this is correct, then the following statements (i.e. the original equations) are true:
745x (mod 689) ≡ 640 (mod 689)
677x (mod 943) ≡ 365 (mod 943)
109x (mod 661) ≡ 957 (mod 661)
Let's see whether that's indeed the case if we use x ≡ 426888466.