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 |
|---|---|---|---|---|---|---|
| 797 | 992 | 0 | 797 | 0 | 1 | 0 |
| 992 | 797 | 1 | 195 | 1 | 0 | 1 |
| 797 | 195 | 4 | 17 | 0 | 1 | -4 |
| 195 | 17 | 11 | 8 | 1 | -4 | 45 |
| 17 | 8 | 2 | 1 | -4 | 45 | -94 |
| 8 | 1 | 8 | 0 | 45 | -94 | 797 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 439 | 227 | 1 | 212 | 0 | 1 | -1 |
| 227 | 212 | 1 | 15 | 1 | -1 | 2 |
| 212 | 15 | 14 | 2 | -1 | 2 | -29 |
| 15 | 2 | 7 | 1 | 2 | -29 | 205 |
| 2 | 1 | 2 | 0 | -29 | 205 | -439 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 647 | 452 | 1 | 195 | 0 | 1 | -1 |
| 452 | 195 | 2 | 62 | 1 | -1 | 3 |
| 195 | 62 | 3 | 9 | -1 | 3 | -10 |
| 62 | 9 | 6 | 8 | 3 | -10 | 63 |
| 9 | 8 | 1 | 1 | -10 | 63 | -73 |
| 8 | 1 | 8 | 0 | 63 | -73 | 647 |
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 ≡ 519 × 992-1 (mod 797) ≡ 519 × 703 (mod 797) ≡ 628 (mod 797)x ≡ 329 × 227-1 (mod 439) ≡ 329 × 205 (mod 439) ≡ 278 (mod 439)x ≡ 764 × 452-1 (mod 647) ≡ 764 × 574 (mod 647) ≡ 517 (mod 647)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 628 (mod 797)
x ≡ 278 (mod 439)
x ≡ 517 (mod 647)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 797 × 439 × 647 = 226374301 - We calculate the numbers M1 to M3
M1=M/m1=226374301/797=284033,M2=M/m2=226374301/439=515659,M3=M/m3=226374301/647=349883 - 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 188 mod 797 ≡ 188n b q r t1 t2 t3 797 284033 0 797 0 1 0 284033 797 356 301 1 0 1 797 301 2 195 0 1 -2 301 195 1 106 1 -2 3 195 106 1 89 -2 3 -5 106 89 1 17 3 -5 8 89 17 5 4 -5 8 -45 17 4 4 1 8 -45 188 4 1 4 0 -45 188 -797
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 119 mod 439 ≡ 119n b q r t1 t2 t3 439 515659 0 439 0 1 0 515659 439 1174 273 1 0 1 439 273 1 166 0 1 -1 273 166 1 107 1 -1 2 166 107 1 59 -1 2 -3 107 59 1 48 2 -3 5 59 48 1 11 -3 5 -8 48 11 4 4 5 -8 37 11 4 2 3 -8 37 -82 4 3 1 1 37 -82 119 3 1 3 0 -82 119 -439
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 319 mod 647 ≡ 319n b q r t1 t2 t3 647 349883 0 647 0 1 0 349883 647 540 503 1 0 1 647 503 1 144 0 1 -1 503 144 3 71 1 -1 4 144 71 2 2 -1 4 -9 71 2 35 1 4 -9 319 2 1 2 0 -9 319 -647
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
= (628 × 284033 × 188 +
278 × 515659 × 119 +
517 × 349883 × 319) mod 226374301
= 89941281 (mod 226374301)
So our answer is 89941281 (mod 226374301).
Verification
So we found that x ≡ 89941281
If this is correct, then the following statements (i.e. the original equations) are true:
992x (mod 797) ≡ 519 (mod 797)
227x (mod 439) ≡ 329 (mod 439)
452x (mod 647) ≡ 764 (mod 647)
Let's see whether that's indeed the case if we use x ≡ 89941281.