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 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 |
|---|---|---|---|---|---|---|
| 521 | 284 | 1 | 237 | 0 | 1 | -1 |
| 284 | 237 | 1 | 47 | 1 | -1 | 2 |
| 237 | 47 | 5 | 2 | -1 | 2 | -11 |
| 47 | 2 | 23 | 1 | 2 | -11 | 255 |
| 2 | 1 | 2 | 0 | -11 | 255 | -521 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 635 | 507 | 1 | 128 | 0 | 1 | -1 |
| 507 | 128 | 3 | 123 | 1 | -1 | 4 |
| 128 | 123 | 1 | 5 | -1 | 4 | -5 |
| 123 | 5 | 24 | 3 | 4 | -5 | 124 |
| 5 | 3 | 1 | 2 | -5 | 124 | -129 |
| 3 | 2 | 1 | 1 | 124 | -129 | 253 |
| 2 | 1 | 2 | 0 | -129 | 253 | -635 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 721 | 198 | 3 | 127 | 0 | 1 | -3 |
| 198 | 127 | 1 | 71 | 1 | -3 | 4 |
| 127 | 71 | 1 | 56 | -3 | 4 | -7 |
| 71 | 56 | 1 | 15 | 4 | -7 | 11 |
| 56 | 15 | 3 | 11 | -7 | 11 | -40 |
| 15 | 11 | 1 | 4 | 11 | -40 | 51 |
| 11 | 4 | 2 | 3 | -40 | 51 | -142 |
| 4 | 3 | 1 | 1 | 51 | -142 | 193 |
| 3 | 1 | 3 | 0 | -142 | 193 | -721 |
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 ≡ 177 × 284-1 (mod 521) ≡ 177 × 255 (mod 521) ≡ 329 (mod 521)x ≡ 408 × 507-1 (mod 635) ≡ 408 × 253 (mod 635) ≡ 354 (mod 635)x ≡ 339 × 198-1 (mod 721) ≡ 339 × 193 (mod 721) ≡ 537 (mod 721)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 329 (mod 521)
x ≡ 354 (mod 635)
x ≡ 537 (mod 721)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 521 × 635 × 721 = 238532035 - We calculate the numbers M1 to M3
M1=M/m1=238532035/521=457835,M2=M/m2=238532035/635=375641,M3=M/m3=238532035/721=330835 - 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 21 mod 521 ≡ 21n b q r t1 t2 t3 521 457835 0 521 0 1 0 457835 521 878 397 1 0 1 521 397 1 124 0 1 -1 397 124 3 25 1 -1 4 124 25 4 24 -1 4 -17 25 24 1 1 4 -17 21 24 1 24 0 -17 21 -521
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 66 mod 635 ≡ 66n b q r t1 t2 t3 635 375641 0 635 0 1 0 375641 635 591 356 1 0 1 635 356 1 279 0 1 -1 356 279 1 77 1 -1 2 279 77 3 48 -1 2 -7 77 48 1 29 2 -7 9 48 29 1 19 -7 9 -16 29 19 1 10 9 -16 25 19 10 1 9 -16 25 -41 10 9 1 1 25 -41 66 9 1 9 0 -41 66 -635
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -104 mod 721 ≡ 617n b q r t1 t2 t3 721 330835 0 721 0 1 0 330835 721 458 617 1 0 1 721 617 1 104 0 1 -1 617 104 5 97 1 -1 6 104 97 1 7 -1 6 -7 97 7 13 6 6 -7 97 7 6 1 1 -7 97 -104 6 1 6 0 97 -104 721
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
= (329 × 457835 × 21 +
354 × 375641 × 66 +
537 × 330835 × 617) mod 238532035
= 142082239 (mod 238532035)
So our answer is 142082239 (mod 238532035).
Verification
So we found that x ≡ 142082239
If this is correct, then the following statements (i.e. the original equations) are true:
284x (mod 521) ≡ 177 (mod 521)
507x (mod 635) ≡ 408 (mod 635)
198x (mod 721) ≡ 339 (mod 721)
Let's see whether that's indeed the case if we use x ≡ 142082239.