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 |
|---|---|---|---|---|---|---|
| 492 | 725 | 0 | 492 | 0 | 1 | 0 |
| 725 | 492 | 1 | 233 | 1 | 0 | 1 |
| 492 | 233 | 2 | 26 | 0 | 1 | -2 |
| 233 | 26 | 8 | 25 | 1 | -2 | 17 |
| 26 | 25 | 1 | 1 | -2 | 17 | -19 |
| 25 | 1 | 25 | 0 | 17 | -19 | 492 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 259 | 261 | 0 | 259 | 0 | 1 | 0 |
| 261 | 259 | 1 | 2 | 1 | 0 | 1 |
| 259 | 2 | 129 | 1 | 0 | 1 | -129 |
| 2 | 1 | 2 | 0 | 1 | -129 | 259 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 613 | 909 | 0 | 613 | 0 | 1 | 0 |
| 909 | 613 | 1 | 296 | 1 | 0 | 1 |
| 613 | 296 | 2 | 21 | 0 | 1 | -2 |
| 296 | 21 | 14 | 2 | 1 | -2 | 29 |
| 21 | 2 | 10 | 1 | -2 | 29 | -292 |
| 2 | 1 | 2 | 0 | 29 | -292 | 613 |
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 ≡ 642 × 725-1 (mod 492) ≡ 642 × 473 (mod 492) ≡ 102 (mod 492)x ≡ 854 × 261-1 (mod 259) ≡ 854 × 130 (mod 259) ≡ 168 (mod 259)x ≡ 946 × 909-1 (mod 613) ≡ 946 × 321 (mod 613) ≡ 231 (mod 613)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 102 (mod 492)
x ≡ 168 (mod 259)
x ≡ 231 (mod 613)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 492 × 259 × 613 = 78113364 - We calculate the numbers M1 to M3
M1=M/m1=78113364/492=158767,M2=M/m2=78113364/259=301596,M3=M/m3=78113364/613=127428 - 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 175 mod 492 ≡ 175n b q r t1 t2 t3 492 158767 0 492 0 1 0 158767 492 322 343 1 0 1 492 343 1 149 0 1 -1 343 149 2 45 1 -1 3 149 45 3 14 -1 3 -10 45 14 3 3 3 -10 33 14 3 4 2 -10 33 -142 3 2 1 1 33 -142 175 2 1 2 0 -142 175 -492
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -41 mod 259 ≡ 218n b q r t1 t2 t3 259 301596 0 259 0 1 0 301596 259 1164 120 1 0 1 259 120 2 19 0 1 -2 120 19 6 6 1 -2 13 19 6 3 1 -2 13 -41 6 1 6 0 13 -41 259
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -121 mod 613 ≡ 492n b q r t1 t2 t3 613 127428 0 613 0 1 0 127428 613 207 537 1 0 1 613 537 1 76 0 1 -1 537 76 7 5 1 -1 8 76 5 15 1 -1 8 -121 5 1 5 0 8 -121 613
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
= (102 × 158767 × 175 +
168 × 301596 × 218 +
231 × 127428 × 492) mod 78113364
= 6938778 (mod 78113364)
So our answer is 6938778 (mod 78113364).
Verification
So we found that x ≡ 6938778
If this is correct, then the following statements (i.e. the original equations) are true:
725x (mod 492) ≡ 642 (mod 492)
261x (mod 259) ≡ 854 (mod 259)
909x (mod 613) ≡ 946 (mod 613)
Let's see whether that's indeed the case if we use x ≡ 6938778.