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 |
|---|---|---|---|---|---|---|
| 127 | 995 | 0 | 127 | 0 | 1 | 0 |
| 995 | 127 | 7 | 106 | 1 | 0 | 1 |
| 127 | 106 | 1 | 21 | 0 | 1 | -1 |
| 106 | 21 | 5 | 1 | 1 | -1 | 6 |
| 21 | 1 | 21 | 0 | -1 | 6 | -127 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 653 | 951 | 0 | 653 | 0 | 1 | 0 |
| 951 | 653 | 1 | 298 | 1 | 0 | 1 |
| 653 | 298 | 2 | 57 | 0 | 1 | -2 |
| 298 | 57 | 5 | 13 | 1 | -2 | 11 |
| 57 | 13 | 4 | 5 | -2 | 11 | -46 |
| 13 | 5 | 2 | 3 | 11 | -46 | 103 |
| 5 | 3 | 1 | 2 | -46 | 103 | -149 |
| 3 | 2 | 1 | 1 | 103 | -149 | 252 |
| 2 | 1 | 2 | 0 | -149 | 252 | -653 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 421 | 656 | 0 | 421 | 0 | 1 | 0 |
| 656 | 421 | 1 | 235 | 1 | 0 | 1 |
| 421 | 235 | 1 | 186 | 0 | 1 | -1 |
| 235 | 186 | 1 | 49 | 1 | -1 | 2 |
| 186 | 49 | 3 | 39 | -1 | 2 | -7 |
| 49 | 39 | 1 | 10 | 2 | -7 | 9 |
| 39 | 10 | 3 | 9 | -7 | 9 | -34 |
| 10 | 9 | 1 | 1 | 9 | -34 | 43 |
| 9 | 1 | 9 | 0 | -34 | 43 | -421 |
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 ≡ 675 × 995-1 (mod 127) ≡ 675 × 6 (mod 127) ≡ 113 (mod 127)x ≡ 250 × 951-1 (mod 653) ≡ 250 × 252 (mod 653) ≡ 312 (mod 653)x ≡ 833 × 656-1 (mod 421) ≡ 833 × 43 (mod 421) ≡ 34 (mod 421)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 113 (mod 127)
x ≡ 312 (mod 653)
x ≡ 34 (mod 421)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 127 × 653 × 421 = 34913951 - We calculate the numbers M1 to M3
M1=M/m1=34913951/127=274913,M2=M/m2=34913951/653=53467,M3=M/m3=34913951/421=82931 - 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 3 mod 127 ≡ 3n b q r t1 t2 t3 127 274913 0 127 0 1 0 274913 127 2164 85 1 0 1 127 85 1 42 0 1 -1 85 42 2 1 1 -1 3 42 1 42 0 -1 3 -127
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -124 mod 653 ≡ 529n b q r t1 t2 t3 653 53467 0 653 0 1 0 53467 653 81 574 1 0 1 653 574 1 79 0 1 -1 574 79 7 21 1 -1 8 79 21 3 16 -1 8 -25 21 16 1 5 8 -25 33 16 5 3 1 -25 33 -124 5 1 5 0 33 -124 653
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 70 mod 421 ≡ 70n b q r t1 t2 t3 421 82931 0 421 0 1 0 82931 421 196 415 1 0 1 421 415 1 6 0 1 -1 415 6 69 1 1 -1 70 6 1 6 0 -1 70 -421
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
= (113 × 274913 × 3 +
312 × 53467 × 529 +
34 × 82931 × 70) mod 34913951
= 2651492 (mod 34913951)
So our answer is 2651492 (mod 34913951).
Verification
So we found that x ≡ 2651492
If this is correct, then the following statements (i.e. the original equations) are true:
995x (mod 127) ≡ 675 (mod 127)
951x (mod 653) ≡ 250 (mod 653)
656x (mod 421) ≡ 833 (mod 421)
Let's see whether that's indeed the case if we use x ≡ 2651492.