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.
This fills all textboxes with random numbers. If you fill in random numbers yourself, it is very likely that those numbers do not have a solution. To avoid disappointment, use this button instead! It only uses random numbers that do have a solution.
This button is similar to the "Clear everything" button, but only clears the left column.
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 |
|---|---|---|---|---|---|---|
| 523 | 394 | 1 | 129 | 0 | 1 | -1 |
| 394 | 129 | 3 | 7 | 1 | -1 | 4 |
| 129 | 7 | 18 | 3 | -1 | 4 | -73 |
| 7 | 3 | 2 | 1 | 4 | -73 | 150 |
| 3 | 1 | 3 | 0 | -73 | 150 | -523 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 497 | 720 | 0 | 497 | 0 | 1 | 0 |
| 720 | 497 | 1 | 223 | 1 | 0 | 1 |
| 497 | 223 | 2 | 51 | 0 | 1 | -2 |
| 223 | 51 | 4 | 19 | 1 | -2 | 9 |
| 51 | 19 | 2 | 13 | -2 | 9 | -20 |
| 19 | 13 | 1 | 6 | 9 | -20 | 29 |
| 13 | 6 | 2 | 1 | -20 | 29 | -78 |
| 6 | 1 | 6 | 0 | 29 | -78 | 497 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 109 | 276 | 0 | 109 | 0 | 1 | 0 |
| 276 | 109 | 2 | 58 | 1 | 0 | 1 |
| 109 | 58 | 1 | 51 | 0 | 1 | -1 |
| 58 | 51 | 1 | 7 | 1 | -1 | 2 |
| 51 | 7 | 7 | 2 | -1 | 2 | -15 |
| 7 | 2 | 3 | 1 | 2 | -15 | 47 |
| 2 | 1 | 2 | 0 | -15 | 47 | -109 |
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 ≡ 232 × 394-1 (mod 523) ≡ 232 × 150 (mod 523) ≡ 282 (mod 523)x ≡ 476 × 720-1 (mod 497) ≡ 476 × 419 (mod 497) ≡ 147 (mod 497)x ≡ 611 × 276-1 (mod 109) ≡ 611 × 47 (mod 109) ≡ 50 (mod 109)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 282 (mod 523)
x ≡ 147 (mod 497)
x ≡ 50 (mod 109)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 523 × 497 × 109 = 28332479 - We calculate the numbers M1 to M3
M1=M/m1=28332479/523=54173,M2=M/m2=28332479/497=57007,M3=M/m3=28332479/109=259931 - 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 160 mod 523 ≡ 160n b q r t1 t2 t3 523 54173 0 523 0 1 0 54173 523 103 304 1 0 1 523 304 1 219 0 1 -1 304 219 1 85 1 -1 2 219 85 2 49 -1 2 -5 85 49 1 36 2 -5 7 49 36 1 13 -5 7 -12 36 13 2 10 7 -12 31 13 10 1 3 -12 31 -43 10 3 3 1 31 -43 160 3 1 3 0 -43 160 -523
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -225 mod 497 ≡ 272n b q r t1 t2 t3 497 57007 0 497 0 1 0 57007 497 114 349 1 0 1 497 349 1 148 0 1 -1 349 148 2 53 1 -1 3 148 53 2 42 -1 3 -7 53 42 1 11 3 -7 10 42 11 3 9 -7 10 -37 11 9 1 2 10 -37 47 9 2 4 1 -37 47 -225 2 1 2 0 47 -225 497
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 16 mod 109 ≡ 16n b q r t1 t2 t3 109 259931 0 109 0 1 0 259931 109 2384 75 1 0 1 109 75 1 34 0 1 -1 75 34 2 7 1 -1 3 34 7 4 6 -1 3 -13 7 6 1 1 3 -13 16 6 1 6 0 -13 16 -109
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
= (282 × 54173 × 160 +
147 × 57007 × 272 +
50 × 259931 × 16) mod 28332479
= 1747102 (mod 28332479)
So our answer is 1747102 (mod 28332479).
Verification
So we found that x ≡ 1747102
If this is correct, then the following statements (i.e. the original equations) are true:
394x (mod 523) ≡ 232 (mod 523)
720x (mod 497) ≡ 476 (mod 497)
276x (mod 109) ≡ 611 (mod 109)
Let's see whether that's indeed the case if we use x ≡ 1747102.