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 |
|---|---|---|---|---|---|---|
| 709 | 114 | 6 | 25 | 0 | 1 | -6 |
| 114 | 25 | 4 | 14 | 1 | -6 | 25 |
| 25 | 14 | 1 | 11 | -6 | 25 | -31 |
| 14 | 11 | 1 | 3 | 25 | -31 | 56 |
| 11 | 3 | 3 | 2 | -31 | 56 | -199 |
| 3 | 2 | 1 | 1 | 56 | -199 | 255 |
| 2 | 1 | 2 | 0 | -199 | 255 | -709 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 601 | 483 | 1 | 118 | 0 | 1 | -1 |
| 483 | 118 | 4 | 11 | 1 | -1 | 5 |
| 118 | 11 | 10 | 8 | -1 | 5 | -51 |
| 11 | 8 | 1 | 3 | 5 | -51 | 56 |
| 8 | 3 | 2 | 2 | -51 | 56 | -163 |
| 3 | 2 | 1 | 1 | 56 | -163 | 219 |
| 2 | 1 | 2 | 0 | -163 | 219 | -601 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 521 | 263 | 1 | 258 | 0 | 1 | -1 |
| 263 | 258 | 1 | 5 | 1 | -1 | 2 |
| 258 | 5 | 51 | 3 | -1 | 2 | -103 |
| 5 | 3 | 1 | 2 | 2 | -103 | 105 |
| 3 | 2 | 1 | 1 | -103 | 105 | -208 |
| 2 | 1 | 2 | 0 | 105 | -208 | 521 |
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 ≡ 868 × 114-1 (mod 709) ≡ 868 × 255 (mod 709) ≡ 132 (mod 709)x ≡ 459 × 483-1 (mod 601) ≡ 459 × 219 (mod 601) ≡ 154 (mod 601)x ≡ 622 × 263-1 (mod 521) ≡ 622 × 313 (mod 521) ≡ 353 (mod 521)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 132 (mod 709)
x ≡ 154 (mod 601)
x ≡ 353 (mod 521)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 709 × 601 × 521 = 222002789 - We calculate the numbers M1 to M3
M1=M/m1=222002789/709=313121,M2=M/m2=222002789/601=369389,M3=M/m3=222002789/521=426109 - 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 80 mod 709 ≡ 80n b q r t1 t2 t3 709 313121 0 709 0 1 0 313121 709 441 452 1 0 1 709 452 1 257 0 1 -1 452 257 1 195 1 -1 2 257 195 1 62 -1 2 -3 195 62 3 9 2 -3 11 62 9 6 8 -3 11 -69 9 8 1 1 11 -69 80 8 1 8 0 -69 80 -709
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -242 mod 601 ≡ 359n b q r t1 t2 t3 601 369389 0 601 0 1 0 369389 601 614 375 1 0 1 601 375 1 226 0 1 -1 375 226 1 149 1 -1 2 226 149 1 77 -1 2 -3 149 77 1 72 2 -3 5 77 72 1 5 -3 5 -8 72 5 14 2 5 -8 117 5 2 2 1 -8 117 -242 2 1 2 0 117 -242 601
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 151 mod 521 ≡ 151n b q r t1 t2 t3 521 426109 0 521 0 1 0 426109 521 817 452 1 0 1 521 452 1 69 0 1 -1 452 69 6 38 1 -1 7 69 38 1 31 -1 7 -8 38 31 1 7 7 -8 15 31 7 4 3 -8 15 -68 7 3 2 1 15 -68 151 3 1 3 0 -68 151 -521
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
= (132 × 313121 × 80 +
154 × 369389 × 359 +
353 × 426109 × 151) mod 222002789
= 42903140 (mod 222002789)
So our answer is 42903140 (mod 222002789).
Verification
So we found that x ≡ 42903140
If this is correct, then the following statements (i.e. the original equations) are true:
114x (mod 709) ≡ 868 (mod 709)
483x (mod 601) ≡ 459 (mod 601)
263x (mod 521) ≡ 622 (mod 521)
Let's see whether that's indeed the case if we use x ≡ 42903140.