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" .
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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 |
|---|---|---|---|---|---|---|
| 527 | 342 | 1 | 185 | 0 | 1 | -1 |
| 342 | 185 | 1 | 157 | 1 | -1 | 2 |
| 185 | 157 | 1 | 28 | -1 | 2 | -3 |
| 157 | 28 | 5 | 17 | 2 | -3 | 17 |
| 28 | 17 | 1 | 11 | -3 | 17 | -20 |
| 17 | 11 | 1 | 6 | 17 | -20 | 37 |
| 11 | 6 | 1 | 5 | -20 | 37 | -57 |
| 6 | 5 | 1 | 1 | 37 | -57 | 94 |
| 5 | 1 | 5 | 0 | -57 | 94 | -527 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 353 | 88 | 4 | 1 | 0 | 1 | -4 |
| 88 | 1 | 88 | 0 | 1 | -4 | 353 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 973 | 188 | 5 | 33 | 0 | 1 | -5 |
| 188 | 33 | 5 | 23 | 1 | -5 | 26 |
| 33 | 23 | 1 | 10 | -5 | 26 | -31 |
| 23 | 10 | 2 | 3 | 26 | -31 | 88 |
| 10 | 3 | 3 | 1 | -31 | 88 | -295 |
| 3 | 1 | 3 | 0 | 88 | -295 | 973 |
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 ≡ 773 × 342-1 (mod 527) ≡ 773 × 94 (mod 527) ≡ 463 (mod 527)x ≡ 865 × 88-1 (mod 353) ≡ 865 × 349 (mod 353) ≡ 70 (mod 353)x ≡ 243 × 188-1 (mod 973) ≡ 243 × 678 (mod 973) ≡ 317 (mod 973)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 463 (mod 527)
x ≡ 70 (mod 353)
x ≡ 317 (mod 973)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 527 × 353 × 973 = 181008163 - We calculate the numbers M1 to M3
M1=M/m1=181008163/527=343469,M2=M/m2=181008163/353=512771,M3=M/m3=181008163/973=186031 - 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 -203 mod 527 ≡ 324n b q r t1 t2 t3 527 343469 0 527 0 1 0 343469 527 651 392 1 0 1 527 392 1 135 0 1 -1 392 135 2 122 1 -1 3 135 122 1 13 -1 3 -4 122 13 9 5 3 -4 39 13 5 2 3 -4 39 -82 5 3 1 2 39 -82 121 3 2 1 1 -82 121 -203 2 1 2 0 121 -203 527
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -110 mod 353 ≡ 243n b q r t1 t2 t3 353 512771 0 353 0 1 0 512771 353 1452 215 1 0 1 353 215 1 138 0 1 -1 215 138 1 77 1 -1 2 138 77 1 61 -1 2 -3 77 61 1 16 2 -3 5 61 16 3 13 -3 5 -18 16 13 1 3 5 -18 23 13 3 4 1 -18 23 -110 3 1 3 0 23 -110 353
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -295 mod 973 ≡ 678n b q r t1 t2 t3 973 186031 0 973 0 1 0 186031 973 191 188 1 0 1 973 188 5 33 0 1 -5 188 33 5 23 1 -5 26 33 23 1 10 -5 26 -31 23 10 2 3 26 -31 88 10 3 3 1 -31 88 -295 3 1 3 0 88 -295 973
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
= (463 × 343469 × 324 +
70 × 512771 × 243 +
317 × 186031 × 678) mod 181008163
= 132090905 (mod 181008163)
So our answer is 132090905 (mod 181008163).
Verification
So we found that x ≡ 132090905
If this is correct, then the following statements (i.e. the original equations) are true:
342x (mod 527) ≡ 773 (mod 527)
88x (mod 353) ≡ 865 (mod 353)
188x (mod 973) ≡ 243 (mod 973)
Let's see whether that's indeed the case if we use x ≡ 132090905.