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 |
|---|---|---|---|---|---|---|
| 581 | 985 | 0 | 581 | 0 | 1 | 0 |
| 985 | 581 | 1 | 404 | 1 | 0 | 1 |
| 581 | 404 | 1 | 177 | 0 | 1 | -1 |
| 404 | 177 | 2 | 50 | 1 | -1 | 3 |
| 177 | 50 | 3 | 27 | -1 | 3 | -10 |
| 50 | 27 | 1 | 23 | 3 | -10 | 13 |
| 27 | 23 | 1 | 4 | -10 | 13 | -23 |
| 23 | 4 | 5 | 3 | 13 | -23 | 128 |
| 4 | 3 | 1 | 1 | -23 | 128 | -151 |
| 3 | 1 | 3 | 0 | 128 | -151 | 581 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 457 | 801 | 0 | 457 | 0 | 1 | 0 |
| 801 | 457 | 1 | 344 | 1 | 0 | 1 |
| 457 | 344 | 1 | 113 | 0 | 1 | -1 |
| 344 | 113 | 3 | 5 | 1 | -1 | 4 |
| 113 | 5 | 22 | 3 | -1 | 4 | -89 |
| 5 | 3 | 1 | 2 | 4 | -89 | 93 |
| 3 | 2 | 1 | 1 | -89 | 93 | -182 |
| 2 | 1 | 2 | 0 | 93 | -182 | 457 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 157 | 30 | 5 | 7 | 0 | 1 | -5 |
| 30 | 7 | 4 | 2 | 1 | -5 | 21 |
| 7 | 2 | 3 | 1 | -5 | 21 | -68 |
| 2 | 1 | 2 | 0 | 21 | -68 | 157 |
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 ≡ 731 × 985-1 (mod 581) ≡ 731 × 430 (mod 581) ≡ 9 (mod 581)x ≡ 503 × 801-1 (mod 457) ≡ 503 × 275 (mod 457) ≡ 311 (mod 457)x ≡ 759 × 30-1 (mod 157) ≡ 759 × 89 (mod 157) ≡ 41 (mod 157)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 9 (mod 581)
x ≡ 311 (mod 457)
x ≡ 41 (mod 157)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 581 × 457 × 157 = 41686169 - We calculate the numbers M1 to M3
M1=M/m1=41686169/581=71749,M2=M/m2=41686169/457=91217,M3=M/m3=41686169/157=265517 - 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 258 mod 581 ≡ 258n b q r t1 t2 t3 581 71749 0 581 0 1 0 71749 581 123 286 1 0 1 581 286 2 9 0 1 -2 286 9 31 7 1 -2 63 9 7 1 2 -2 63 -65 7 2 3 1 63 -65 258 2 1 2 0 -65 258 -581
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -5 mod 457 ≡ 452n b q r t1 t2 t3 457 91217 0 457 0 1 0 91217 457 199 274 1 0 1 457 274 1 183 0 1 -1 274 183 1 91 1 -1 2 183 91 2 1 -1 2 -5 91 1 91 0 2 -5 457
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -68 mod 157 ≡ 89n b q r t1 t2 t3 157 265517 0 157 0 1 0 265517 157 1691 30 1 0 1 157 30 5 7 0 1 -5 30 7 4 2 1 -5 21 7 2 3 1 -5 21 -68 2 1 2 0 21 -68 157
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
= (9 × 71749 × 258 +
311 × 91217 × 452 +
41 × 265517 × 89) mod 41686169
= 34848389 (mod 41686169)
So our answer is 34848389 (mod 41686169).
Verification
So we found that x ≡ 34848389
If this is correct, then the following statements (i.e. the original equations) are true:
985x (mod 581) ≡ 731 (mod 581)
801x (mod 457) ≡ 503 (mod 457)
30x (mod 157) ≡ 759 (mod 157)
Let's see whether that's indeed the case if we use x ≡ 34848389.