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 |
|---|---|---|---|---|---|---|
| 503 | 730 | 0 | 503 | 0 | 1 | 0 |
| 730 | 503 | 1 | 227 | 1 | 0 | 1 |
| 503 | 227 | 2 | 49 | 0 | 1 | -2 |
| 227 | 49 | 4 | 31 | 1 | -2 | 9 |
| 49 | 31 | 1 | 18 | -2 | 9 | -11 |
| 31 | 18 | 1 | 13 | 9 | -11 | 20 |
| 18 | 13 | 1 | 5 | -11 | 20 | -31 |
| 13 | 5 | 2 | 3 | 20 | -31 | 82 |
| 5 | 3 | 1 | 2 | -31 | 82 | -113 |
| 3 | 2 | 1 | 1 | 82 | -113 | 195 |
| 2 | 1 | 2 | 0 | -113 | 195 | -503 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 491 | 287 | 1 | 204 | 0 | 1 | -1 |
| 287 | 204 | 1 | 83 | 1 | -1 | 2 |
| 204 | 83 | 2 | 38 | -1 | 2 | -5 |
| 83 | 38 | 2 | 7 | 2 | -5 | 12 |
| 38 | 7 | 5 | 3 | -5 | 12 | -65 |
| 7 | 3 | 2 | 1 | 12 | -65 | 142 |
| 3 | 1 | 3 | 0 | -65 | 142 | -491 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 953 | 155 | 6 | 23 | 0 | 1 | -6 |
| 155 | 23 | 6 | 17 | 1 | -6 | 37 |
| 23 | 17 | 1 | 6 | -6 | 37 | -43 |
| 17 | 6 | 2 | 5 | 37 | -43 | 123 |
| 6 | 5 | 1 | 1 | -43 | 123 | -166 |
| 5 | 1 | 5 | 0 | 123 | -166 | 953 |
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 ≡ 732 × 730-1 (mod 503) ≡ 732 × 195 (mod 503) ≡ 391 (mod 503)x ≡ 498 × 287-1 (mod 491) ≡ 498 × 142 (mod 491) ≡ 12 (mod 491)x ≡ 111 × 155-1 (mod 953) ≡ 111 × 787 (mod 953) ≡ 634 (mod 953)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 391 (mod 503)
x ≡ 12 (mod 491)
x ≡ 634 (mod 953)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 503 × 491 × 953 = 235365269 - We calculate the numbers M1 to M3
M1=M/m1=235365269/503=467923,M2=M/m2=235365269/491=479359,M3=M/m3=235365269/953=246973 - 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 -208 mod 503 ≡ 295n b q r t1 t2 t3 503 467923 0 503 0 1 0 467923 503 930 133 1 0 1 503 133 3 104 0 1 -3 133 104 1 29 1 -3 4 104 29 3 17 -3 4 -15 29 17 1 12 4 -15 19 17 12 1 5 -15 19 -34 12 5 2 2 19 -34 87 5 2 2 1 -34 87 -208 2 1 2 0 87 -208 503
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -103 mod 491 ≡ 388n b q r t1 t2 t3 491 479359 0 491 0 1 0 479359 491 976 143 1 0 1 491 143 3 62 0 1 -3 143 62 2 19 1 -3 7 62 19 3 5 -3 7 -24 19 5 3 4 7 -24 79 5 4 1 1 -24 79 -103 4 1 4 0 79 -103 491
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -359 mod 953 ≡ 594n b q r t1 t2 t3 953 246973 0 953 0 1 0 246973 953 259 146 1 0 1 953 146 6 77 0 1 -6 146 77 1 69 1 -6 7 77 69 1 8 -6 7 -13 69 8 8 5 7 -13 111 8 5 1 3 -13 111 -124 5 3 1 2 111 -124 235 3 2 1 1 -124 235 -359 2 1 2 0 235 -359 953
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
= (391 × 467923 × 295 +
12 × 479359 × 388 +
634 × 246973 × 594) mod 235365269
= 227302570 (mod 235365269)
So our answer is 227302570 (mod 235365269).
Verification
So we found that x ≡ 227302570
If this is correct, then the following statements (i.e. the original equations) are true:
730x (mod 503) ≡ 732 (mod 503)
287x (mod 491) ≡ 498 (mod 491)
155x (mod 953) ≡ 111 (mod 953)
Let's see whether that's indeed the case if we use x ≡ 227302570.