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 |
|---|---|---|---|---|---|---|
| 507 | 131 | 3 | 114 | 0 | 1 | -3 |
| 131 | 114 | 1 | 17 | 1 | -3 | 4 |
| 114 | 17 | 6 | 12 | -3 | 4 | -27 |
| 17 | 12 | 1 | 5 | 4 | -27 | 31 |
| 12 | 5 | 2 | 2 | -27 | 31 | -89 |
| 5 | 2 | 2 | 1 | 31 | -89 | 209 |
| 2 | 1 | 2 | 0 | -89 | 209 | -507 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 251 | 173 | 1 | 78 | 0 | 1 | -1 |
| 173 | 78 | 2 | 17 | 1 | -1 | 3 |
| 78 | 17 | 4 | 10 | -1 | 3 | -13 |
| 17 | 10 | 1 | 7 | 3 | -13 | 16 |
| 10 | 7 | 1 | 3 | -13 | 16 | -29 |
| 7 | 3 | 2 | 1 | 16 | -29 | 74 |
| 3 | 1 | 3 | 0 | -29 | 74 | -251 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 655 | 773 | 0 | 655 | 0 | 1 | 0 |
| 773 | 655 | 1 | 118 | 1 | 0 | 1 |
| 655 | 118 | 5 | 65 | 0 | 1 | -5 |
| 118 | 65 | 1 | 53 | 1 | -5 | 6 |
| 65 | 53 | 1 | 12 | -5 | 6 | -11 |
| 53 | 12 | 4 | 5 | 6 | -11 | 50 |
| 12 | 5 | 2 | 2 | -11 | 50 | -111 |
| 5 | 2 | 2 | 1 | 50 | -111 | 272 |
| 2 | 1 | 2 | 0 | -111 | 272 | -655 |
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 ≡ 933 × 131-1 (mod 507) ≡ 933 × 209 (mod 507) ≡ 309 (mod 507)x ≡ 233 × 173-1 (mod 251) ≡ 233 × 74 (mod 251) ≡ 174 (mod 251)x ≡ 339 × 773-1 (mod 655) ≡ 339 × 272 (mod 655) ≡ 508 (mod 655)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 309 (mod 507)
x ≡ 174 (mod 251)
x ≡ 508 (mod 655)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 507 × 251 × 655 = 83353335 - We calculate the numbers M1 to M3
M1=M/m1=83353335/507=164405,M2=M/m2=83353335/251=332085,M3=M/m3=83353335/655=127257 - 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 -37 mod 507 ≡ 470n b q r t1 t2 t3 507 164405 0 507 0 1 0 164405 507 324 137 1 0 1 507 137 3 96 0 1 -3 137 96 1 41 1 -3 4 96 41 2 14 -3 4 -11 41 14 2 13 4 -11 26 14 13 1 1 -11 26 -37 13 1 13 0 26 -37 507
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 21 mod 251 ≡ 21n b q r t1 t2 t3 251 332085 0 251 0 1 0 332085 251 1323 12 1 0 1 251 12 20 11 0 1 -20 12 11 1 1 1 -20 21 11 1 11 0 -20 21 -251
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -7 mod 655 ≡ 648n b q r t1 t2 t3 655 127257 0 655 0 1 0 127257 655 194 187 1 0 1 655 187 3 94 0 1 -3 187 94 1 93 1 -3 4 94 93 1 1 -3 4 -7 93 1 93 0 4 -7 655
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
= (309 × 164405 × 470 +
174 × 332085 × 21 +
508 × 127257 × 648) mod 83353335
= 48217023 (mod 83353335)
So our answer is 48217023 (mod 83353335).
Verification
So we found that x ≡ 48217023
If this is correct, then the following statements (i.e. the original equations) are true:
131x (mod 507) ≡ 933 (mod 507)
173x (mod 251) ≡ 233 (mod 251)
773x (mod 655) ≡ 339 (mod 655)
Let's see whether that's indeed the case if we use x ≡ 48217023.