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 |
|---|---|---|---|---|---|---|
| 179 | 535 | 0 | 179 | 0 | 1 | 0 |
| 535 | 179 | 2 | 177 | 1 | 0 | 1 |
| 179 | 177 | 1 | 2 | 0 | 1 | -1 |
| 177 | 2 | 88 | 1 | 1 | -1 | 89 |
| 2 | 1 | 2 | 0 | -1 | 89 | -179 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 253 | 94 | 2 | 65 | 0 | 1 | -2 |
| 94 | 65 | 1 | 29 | 1 | -2 | 3 |
| 65 | 29 | 2 | 7 | -2 | 3 | -8 |
| 29 | 7 | 4 | 1 | 3 | -8 | 35 |
| 7 | 1 | 7 | 0 | -8 | 35 | -253 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 691 | 945 | 0 | 691 | 0 | 1 | 0 |
| 945 | 691 | 1 | 254 | 1 | 0 | 1 |
| 691 | 254 | 2 | 183 | 0 | 1 | -2 |
| 254 | 183 | 1 | 71 | 1 | -2 | 3 |
| 183 | 71 | 2 | 41 | -2 | 3 | -8 |
| 71 | 41 | 1 | 30 | 3 | -8 | 11 |
| 41 | 30 | 1 | 11 | -8 | 11 | -19 |
| 30 | 11 | 2 | 8 | 11 | -19 | 49 |
| 11 | 8 | 1 | 3 | -19 | 49 | -68 |
| 8 | 3 | 2 | 2 | 49 | -68 | 185 |
| 3 | 2 | 1 | 1 | -68 | 185 | -253 |
| 2 | 1 | 2 | 0 | 185 | -253 | 691 |
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 ≡ 451 × 535-1 (mod 179) ≡ 451 × 89 (mod 179) ≡ 43 (mod 179)x ≡ 320 × 94-1 (mod 253) ≡ 320 × 35 (mod 253) ≡ 68 (mod 253)x ≡ 391 × 945-1 (mod 691) ≡ 391 × 438 (mod 691) ≡ 581 (mod 691)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 43 (mod 179)
x ≡ 68 (mod 253)
x ≡ 581 (mod 691)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 179 × 253 × 691 = 31293317 - We calculate the numbers M1 to M3
M1=M/m1=31293317/179=174823,M2=M/m2=31293317/253=123689,M3=M/m3=31293317/691=45287 - 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 179 ≡ 176n b q r t1 t2 t3 179 174823 0 179 0 1 0 174823 179 976 119 1 0 1 179 119 1 60 0 1 -1 119 60 1 59 1 -1 2 60 59 1 1 -1 2 -3 59 1 59 0 2 -3 179
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 9 mod 253 ≡ 9n b q r t1 t2 t3 253 123689 0 253 0 1 0 123689 253 488 225 1 0 1 253 225 1 28 0 1 -1 225 28 8 1 1 -1 9 28 1 28 0 -1 9 -253
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -13 mod 691 ≡ 678n b q r t1 t2 t3 691 45287 0 691 0 1 0 45287 691 65 372 1 0 1 691 372 1 319 0 1 -1 372 319 1 53 1 -1 2 319 53 6 1 -1 2 -13 53 1 53 0 2 -13 691
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
= (43 × 174823 × 176 +
68 × 123689 × 9 +
581 × 45287 × 678) mod 31293317
= 24025960 (mod 31293317)
So our answer is 24025960 (mod 31293317).
Verification
So we found that x ≡ 24025960
If this is correct, then the following statements (i.e. the original equations) are true:
535x (mod 179) ≡ 451 (mod 179)
94x (mod 253) ≡ 320 (mod 253)
945x (mod 691) ≡ 391 (mod 691)
Let's see whether that's indeed the case if we use x ≡ 24025960.