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 |
|---|---|---|---|---|---|---|
| 361 | 291 | 1 | 70 | 0 | 1 | -1 |
| 291 | 70 | 4 | 11 | 1 | -1 | 5 |
| 70 | 11 | 6 | 4 | -1 | 5 | -31 |
| 11 | 4 | 2 | 3 | 5 | -31 | 67 |
| 4 | 3 | 1 | 1 | -31 | 67 | -98 |
| 3 | 1 | 3 | 0 | 67 | -98 | 361 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 651 | 985 | 0 | 651 | 0 | 1 | 0 |
| 985 | 651 | 1 | 334 | 1 | 0 | 1 |
| 651 | 334 | 1 | 317 | 0 | 1 | -1 |
| 334 | 317 | 1 | 17 | 1 | -1 | 2 |
| 317 | 17 | 18 | 11 | -1 | 2 | -37 |
| 17 | 11 | 1 | 6 | 2 | -37 | 39 |
| 11 | 6 | 1 | 5 | -37 | 39 | -76 |
| 6 | 5 | 1 | 1 | 39 | -76 | 115 |
| 5 | 1 | 5 | 0 | -76 | 115 | -651 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 461 | 350 | 1 | 111 | 0 | 1 | -1 |
| 350 | 111 | 3 | 17 | 1 | -1 | 4 |
| 111 | 17 | 6 | 9 | -1 | 4 | -25 |
| 17 | 9 | 1 | 8 | 4 | -25 | 29 |
| 9 | 8 | 1 | 1 | -25 | 29 | -54 |
| 8 | 1 | 8 | 0 | 29 | -54 | 461 |
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 ≡ 572 × 291-1 (mod 361) ≡ 572 × 263 (mod 361) ≡ 260 (mod 361)x ≡ 680 × 985-1 (mod 651) ≡ 680 × 115 (mod 651) ≡ 80 (mod 651)x ≡ 405 × 350-1 (mod 461) ≡ 405 × 407 (mod 461) ≡ 258 (mod 461)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 260 (mod 361)
x ≡ 80 (mod 651)
x ≡ 258 (mod 461)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 361 × 651 × 461 = 108340071 - We calculate the numbers M1 to M3
M1=M/m1=108340071/361=300111,M2=M/m2=108340071/651=166421,M3=M/m3=108340071/461=235011 - 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 361 ≡ 358n b q r t1 t2 t3 361 300111 0 361 0 1 0 300111 361 831 120 1 0 1 361 120 3 1 0 1 -3 120 1 120 0 1 -3 361
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -205 mod 651 ≡ 446n b q r t1 t2 t3 651 166421 0 651 0 1 0 166421 651 255 416 1 0 1 651 416 1 235 0 1 -1 416 235 1 181 1 -1 2 235 181 1 54 -1 2 -3 181 54 3 19 2 -3 11 54 19 2 16 -3 11 -25 19 16 1 3 11 -25 36 16 3 5 1 -25 36 -205 3 1 3 0 36 -205 651
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 149 mod 461 ≡ 149n b q r t1 t2 t3 461 235011 0 461 0 1 0 235011 461 509 362 1 0 1 461 362 1 99 0 1 -1 362 99 3 65 1 -1 4 99 65 1 34 -1 4 -5 65 34 1 31 4 -5 9 34 31 1 3 -5 9 -14 31 3 10 1 9 -14 149 3 1 3 0 -14 149 -461
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
= (260 × 300111 × 358 +
80 × 166421 × 446 +
258 × 235011 × 149) mod 108340071
= 3857906 (mod 108340071)
So our answer is 3857906 (mod 108340071).
Verification
So we found that x ≡ 3857906
If this is correct, then the following statements (i.e. the original equations) are true:
291x (mod 361) ≡ 572 (mod 361)
985x (mod 651) ≡ 680 (mod 651)
350x (mod 461) ≡ 405 (mod 461)
Let's see whether that's indeed the case if we use x ≡ 3857906.