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 |
|---|---|---|---|---|---|---|
| 821 | 683 | 1 | 138 | 0 | 1 | -1 |
| 683 | 138 | 4 | 131 | 1 | -1 | 5 |
| 138 | 131 | 1 | 7 | -1 | 5 | -6 |
| 131 | 7 | 18 | 5 | 5 | -6 | 113 |
| 7 | 5 | 1 | 2 | -6 | 113 | -119 |
| 5 | 2 | 2 | 1 | 113 | -119 | 351 |
| 2 | 1 | 2 | 0 | -119 | 351 | -821 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 941 | 556 | 1 | 385 | 0 | 1 | -1 |
| 556 | 385 | 1 | 171 | 1 | -1 | 2 |
| 385 | 171 | 2 | 43 | -1 | 2 | -5 |
| 171 | 43 | 3 | 42 | 2 | -5 | 17 |
| 43 | 42 | 1 | 1 | -5 | 17 | -22 |
| 42 | 1 | 42 | 0 | 17 | -22 | 941 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 353 | 530 | 0 | 353 | 0 | 1 | 0 |
| 530 | 353 | 1 | 177 | 1 | 0 | 1 |
| 353 | 177 | 1 | 176 | 0 | 1 | -1 |
| 177 | 176 | 1 | 1 | 1 | -1 | 2 |
| 176 | 1 | 176 | 0 | -1 | 2 | -353 |
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 ≡ 963 × 683-1 (mod 821) ≡ 963 × 351 (mod 821) ≡ 582 (mod 821)x ≡ 190 × 556-1 (mod 941) ≡ 190 × 919 (mod 941) ≡ 525 (mod 941)x ≡ 280 × 530-1 (mod 353) ≡ 280 × 2 (mod 353) ≡ 207 (mod 353)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 582 (mod 821)
x ≡ 525 (mod 941)
x ≡ 207 (mod 353)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 821 × 941 × 353 = 272714033 - We calculate the numbers M1 to M3
M1=M/m1=272714033/821=332173,M2=M/m2=272714033/941=289813,M3=M/m3=272714033/353=772561 - 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 319 mod 821 ≡ 319n b q r t1 t2 t3 821 332173 0 821 0 1 0 332173 821 404 489 1 0 1 821 489 1 332 0 1 -1 489 332 1 157 1 -1 2 332 157 2 18 -1 2 -5 157 18 8 13 2 -5 42 18 13 1 5 -5 42 -47 13 5 2 3 42 -47 136 5 3 1 2 -47 136 -183 3 2 1 1 136 -183 319 2 1 2 0 -183 319 -821
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -251 mod 941 ≡ 690n b q r t1 t2 t3 941 289813 0 941 0 1 0 289813 941 307 926 1 0 1 941 926 1 15 0 1 -1 926 15 61 11 1 -1 62 15 11 1 4 -1 62 -63 11 4 2 3 62 -63 188 4 3 1 1 -63 188 -251 3 1 3 0 188 -251 941
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -43 mod 353 ≡ 310n b q r t1 t2 t3 353 772561 0 353 0 1 0 772561 353 2188 197 1 0 1 353 197 1 156 0 1 -1 197 156 1 41 1 -1 2 156 41 3 33 -1 2 -7 41 33 1 8 2 -7 9 33 8 4 1 -7 9 -43 8 1 8 0 9 -43 353
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
= (582 × 332173 × 319 +
525 × 289813 × 690 +
207 × 772561 × 310) mod 272714033
= 241059318 (mod 272714033)
So our answer is 241059318 (mod 272714033).
Verification
So we found that x ≡ 241059318
If this is correct, then the following statements (i.e. the original equations) are true:
683x (mod 821) ≡ 963 (mod 821)
556x (mod 941) ≡ 190 (mod 941)
530x (mod 353) ≡ 280 (mod 353)
Let's see whether that's indeed the case if we use x ≡ 241059318.