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.
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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.
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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 |
|---|---|---|---|---|---|---|
| 379 | 61 | 6 | 13 | 0 | 1 | -6 |
| 61 | 13 | 4 | 9 | 1 | -6 | 25 |
| 13 | 9 | 1 | 4 | -6 | 25 | -31 |
| 9 | 4 | 2 | 1 | 25 | -31 | 87 |
| 4 | 1 | 4 | 0 | -31 | 87 | -379 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 689 | 735 | 0 | 689 | 0 | 1 | 0 |
| 735 | 689 | 1 | 46 | 1 | 0 | 1 |
| 689 | 46 | 14 | 45 | 0 | 1 | -14 |
| 46 | 45 | 1 | 1 | 1 | -14 | 15 |
| 45 | 1 | 45 | 0 | -14 | 15 | -689 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 73 | 852 | 0 | 73 | 0 | 1 | 0 |
| 852 | 73 | 11 | 49 | 1 | 0 | 1 |
| 73 | 49 | 1 | 24 | 0 | 1 | -1 |
| 49 | 24 | 2 | 1 | 1 | -1 | 3 |
| 24 | 1 | 24 | 0 | -1 | 3 | -73 |
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 ≡ 236 × 61-1 (mod 379) ≡ 236 × 87 (mod 379) ≡ 66 (mod 379)x ≡ 88 × 735-1 (mod 689) ≡ 88 × 15 (mod 689) ≡ 631 (mod 689)x ≡ 938 × 852-1 (mod 73) ≡ 938 × 3 (mod 73) ≡ 40 (mod 73)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 66 (mod 379)
x ≡ 631 (mod 689)
x ≡ 40 (mod 73)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 379 × 689 × 73 = 19062563 - We calculate the numbers M1 to M3
M1=M/m1=19062563/379=50297,M2=M/m2=19062563/689=27667,M3=M/m3=19062563/73=261131 - 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 31 mod 379 ≡ 31n b q r t1 t2 t3 379 50297 0 379 0 1 0 50297 379 132 269 1 0 1 379 269 1 110 0 1 -1 269 110 2 49 1 -1 3 110 49 2 12 -1 3 -7 49 12 4 1 3 -7 31 12 1 12 0 -7 31 -379
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -264 mod 689 ≡ 425n b q r t1 t2 t3 689 27667 0 689 0 1 0 27667 689 40 107 1 0 1 689 107 6 47 0 1 -6 107 47 2 13 1 -6 13 47 13 3 8 -6 13 -45 13 8 1 5 13 -45 58 8 5 1 3 -45 58 -103 5 3 1 2 58 -103 161 3 2 1 1 -103 161 -264 2 1 2 0 161 -264 689
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 22 mod 73 ≡ 22n b q r t1 t2 t3 73 261131 0 73 0 1 0 261131 73 3577 10 1 0 1 73 10 7 3 0 1 -7 10 3 3 1 1 -7 22 3 1 3 0 -7 22 -73
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
= (66 × 50297 × 31 +
631 × 27667 × 425 +
40 × 261131 × 22) mod 19062563
= 12900089 (mod 19062563)
So our answer is 12900089 (mod 19062563).
Verification
So we found that x ≡ 12900089
If this is correct, then the following statements (i.e. the original equations) are true:
61x (mod 379) ≡ 236 (mod 379)
735x (mod 689) ≡ 88 (mod 689)
852x (mod 73) ≡ 938 (mod 73)
Let's see whether that's indeed the case if we use x ≡ 12900089.