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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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 |
|---|---|---|---|---|---|---|
| 925 | 588 | 1 | 337 | 0 | 1 | -1 |
| 588 | 337 | 1 | 251 | 1 | -1 | 2 |
| 337 | 251 | 1 | 86 | -1 | 2 | -3 |
| 251 | 86 | 2 | 79 | 2 | -3 | 8 |
| 86 | 79 | 1 | 7 | -3 | 8 | -11 |
| 79 | 7 | 11 | 2 | 8 | -11 | 129 |
| 7 | 2 | 3 | 1 | -11 | 129 | -398 |
| 2 | 1 | 2 | 0 | 129 | -398 | 925 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 376 | 533 | 0 | 376 | 0 | 1 | 0 |
| 533 | 376 | 1 | 157 | 1 | 0 | 1 |
| 376 | 157 | 2 | 62 | 0 | 1 | -2 |
| 157 | 62 | 2 | 33 | 1 | -2 | 5 |
| 62 | 33 | 1 | 29 | -2 | 5 | -7 |
| 33 | 29 | 1 | 4 | 5 | -7 | 12 |
| 29 | 4 | 7 | 1 | -7 | 12 | -91 |
| 4 | 1 | 4 | 0 | 12 | -91 | 376 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 921 | 995 | 0 | 921 | 0 | 1 | 0 |
| 995 | 921 | 1 | 74 | 1 | 0 | 1 |
| 921 | 74 | 12 | 33 | 0 | 1 | -12 |
| 74 | 33 | 2 | 8 | 1 | -12 | 25 |
| 33 | 8 | 4 | 1 | -12 | 25 | -112 |
| 8 | 1 | 8 | 0 | 25 | -112 | 921 |
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 ≡ 543 × 588-1 (mod 925) ≡ 543 × 527 (mod 925) ≡ 336 (mod 925)x ≡ 38 × 533-1 (mod 376) ≡ 38 × 285 (mod 376) ≡ 302 (mod 376)x ≡ 766 × 995-1 (mod 921) ≡ 766 × 809 (mod 921) ≡ 782 (mod 921)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 336 (mod 925)
x ≡ 302 (mod 376)
x ≡ 782 (mod 921)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 925 × 376 × 921 = 320323800 - We calculate the numbers M1 to M3
M1=M/m1=320323800/925=346296,M2=M/m2=320323800/376=851925,M3=M/m3=320323800/921=347800 - 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 131 mod 925 ≡ 131n b q r t1 t2 t3 925 346296 0 925 0 1 0 346296 925 374 346 1 0 1 925 346 2 233 0 1 -2 346 233 1 113 1 -2 3 233 113 2 7 -2 3 -8 113 7 16 1 3 -8 131 7 1 7 0 -8 131 -925
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 157 mod 376 ≡ 157n b q r t1 t2 t3 376 851925 0 376 0 1 0 851925 376 2265 285 1 0 1 376 285 1 91 0 1 -1 285 91 3 12 1 -1 4 91 12 7 7 -1 4 -29 12 7 1 5 4 -29 33 7 5 1 2 -29 33 -62 5 2 2 1 33 -62 157 2 1 2 0 -62 157 -376
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 406 mod 921 ≡ 406n b q r t1 t2 t3 921 347800 0 921 0 1 0 347800 921 377 583 1 0 1 921 583 1 338 0 1 -1 583 338 1 245 1 -1 2 338 245 1 93 -1 2 -3 245 93 2 59 2 -3 8 93 59 1 34 -3 8 -11 59 34 1 25 8 -11 19 34 25 1 9 -11 19 -30 25 9 2 7 19 -30 79 9 7 1 2 -30 79 -109 7 2 3 1 79 -109 406 2 1 2 0 -109 406 -921
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
= (336 × 346296 × 131 +
302 × 851925 × 157 +
782 × 347800 × 406) mod 320323800
= 131725886 (mod 320323800)
So our answer is 131725886 (mod 320323800).
Verification
So we found that x ≡ 131725886
If this is correct, then the following statements (i.e. the original equations) are true:
588x (mod 925) ≡ 543 (mod 925)
533x (mod 376) ≡ 38 (mod 376)
995x (mod 921) ≡ 766 (mod 921)
Let's see whether that's indeed the case if we use x ≡ 131725886.