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 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 |
|---|---|---|---|---|---|---|
| 601 | 247 | 2 | 107 | 0 | 1 | -2 |
| 247 | 107 | 2 | 33 | 1 | -2 | 5 |
| 107 | 33 | 3 | 8 | -2 | 5 | -17 |
| 33 | 8 | 4 | 1 | 5 | -17 | 73 |
| 8 | 1 | 8 | 0 | -17 | 73 | -601 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 201 | 239 | 0 | 201 | 0 | 1 | 0 |
| 239 | 201 | 1 | 38 | 1 | 0 | 1 |
| 201 | 38 | 5 | 11 | 0 | 1 | -5 |
| 38 | 11 | 3 | 5 | 1 | -5 | 16 |
| 11 | 5 | 2 | 1 | -5 | 16 | -37 |
| 5 | 1 | 5 | 0 | 16 | -37 | 201 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 493 | 42 | 11 | 31 | 0 | 1 | -11 |
| 42 | 31 | 1 | 11 | 1 | -11 | 12 |
| 31 | 11 | 2 | 9 | -11 | 12 | -35 |
| 11 | 9 | 1 | 2 | 12 | -35 | 47 |
| 9 | 2 | 4 | 1 | -35 | 47 | -223 |
| 2 | 1 | 2 | 0 | 47 | -223 | 493 |
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 ≡ 390 × 247-1 (mod 601) ≡ 390 × 73 (mod 601) ≡ 223 (mod 601)x ≡ 609 × 239-1 (mod 201) ≡ 609 × 164 (mod 201) ≡ 180 (mod 201)x ≡ 102 × 42-1 (mod 493) ≡ 102 × 270 (mod 493) ≡ 425 (mod 493)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 223 (mod 601)
x ≡ 180 (mod 201)
x ≡ 425 (mod 493)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 601 × 201 × 493 = 59554893 - We calculate the numbers M1 to M3
M1=M/m1=59554893/601=99093,M2=M/m2=59554893/201=296293,M3=M/m3=59554893/493=120801 - 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 -192 mod 601 ≡ 409n b q r t1 t2 t3 601 99093 0 601 0 1 0 99093 601 164 529 1 0 1 601 529 1 72 0 1 -1 529 72 7 25 1 -1 8 72 25 2 22 -1 8 -17 25 22 1 3 8 -17 25 22 3 7 1 -17 25 -192 3 1 3 0 25 -192 601
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -74 mod 201 ≡ 127n b q r t1 t2 t3 201 296293 0 201 0 1 0 296293 201 1474 19 1 0 1 201 19 10 11 0 1 -10 19 11 1 8 1 -10 11 11 8 1 3 -10 11 -21 8 3 2 2 11 -21 53 3 2 1 1 -21 53 -74 2 1 2 0 53 -74 201
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -154 mod 493 ≡ 339n b q r t1 t2 t3 493 120801 0 493 0 1 0 120801 493 245 16 1 0 1 493 16 30 13 0 1 -30 16 13 1 3 1 -30 31 13 3 4 1 -30 31 -154 3 1 3 0 31 -154 493
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
= (223 × 99093 × 409 +
180 × 296293 × 127 +
425 × 120801 × 339) mod 59554893
= 43561905 (mod 59554893)
So our answer is 43561905 (mod 59554893).
Verification
So we found that x ≡ 43561905
If this is correct, then the following statements (i.e. the original equations) are true:
247x (mod 601) ≡ 390 (mod 601)
239x (mod 201) ≡ 609 (mod 201)
42x (mod 493) ≡ 102 (mod 493)
Let's see whether that's indeed the case if we use x ≡ 43561905.