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 |
|---|---|---|---|---|---|---|
| 481 | 238 | 2 | 5 | 0 | 1 | -2 |
| 238 | 5 | 47 | 3 | 1 | -2 | 95 |
| 5 | 3 | 1 | 2 | -2 | 95 | -97 |
| 3 | 2 | 1 | 1 | 95 | -97 | 192 |
| 2 | 1 | 2 | 0 | -97 | 192 | -481 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 223 | 211 | 1 | 12 | 0 | 1 | -1 |
| 211 | 12 | 17 | 7 | 1 | -1 | 18 |
| 12 | 7 | 1 | 5 | -1 | 18 | -19 |
| 7 | 5 | 1 | 2 | 18 | -19 | 37 |
| 5 | 2 | 2 | 1 | -19 | 37 | -93 |
| 2 | 1 | 2 | 0 | 37 | -93 | 223 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 469 | 580 | 0 | 469 | 0 | 1 | 0 |
| 580 | 469 | 1 | 111 | 1 | 0 | 1 |
| 469 | 111 | 4 | 25 | 0 | 1 | -4 |
| 111 | 25 | 4 | 11 | 1 | -4 | 17 |
| 25 | 11 | 2 | 3 | -4 | 17 | -38 |
| 11 | 3 | 3 | 2 | 17 | -38 | 131 |
| 3 | 2 | 1 | 1 | -38 | 131 | -169 |
| 2 | 1 | 2 | 0 | 131 | -169 | 469 |
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 ≡ 726 × 238-1 (mod 481) ≡ 726 × 192 (mod 481) ≡ 383 (mod 481)x ≡ 244 × 211-1 (mod 223) ≡ 244 × 130 (mod 223) ≡ 54 (mod 223)x ≡ 504 × 580-1 (mod 469) ≡ 504 × 300 (mod 469) ≡ 182 (mod 469)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 383 (mod 481)
x ≡ 54 (mod 223)
x ≡ 182 (mod 469)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 481 × 223 × 469 = 50306347 - We calculate the numbers M1 to M3
M1=M/m1=50306347/481=104587,M2=M/m2=50306347/223=225589,M3=M/m3=50306347/469=107263 - 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 -71 mod 481 ≡ 410n b q r t1 t2 t3 481 104587 0 481 0 1 0 104587 481 217 210 1 0 1 481 210 2 61 0 1 -2 210 61 3 27 1 -2 7 61 27 2 7 -2 7 -16 27 7 3 6 7 -16 55 7 6 1 1 -16 55 -71 6 1 6 0 55 -71 481
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 41 mod 223 ≡ 41n b q r t1 t2 t3 223 225589 0 223 0 1 0 225589 223 1011 136 1 0 1 223 136 1 87 0 1 -1 136 87 1 49 1 -1 2 87 49 1 38 -1 2 -3 49 38 1 11 2 -3 5 38 11 3 5 -3 5 -18 11 5 2 1 5 -18 41 5 1 5 0 -18 41 -223
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -17 mod 469 ≡ 452n b q r t1 t2 t3 469 107263 0 469 0 1 0 107263 469 228 331 1 0 1 469 331 1 138 0 1 -1 331 138 2 55 1 -1 3 138 55 2 28 -1 3 -7 55 28 1 27 3 -7 10 28 27 1 1 -7 10 -17 27 1 27 0 10 -17 469
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
= (383 × 104587 × 410 +
54 × 225589 × 41 +
182 × 107263 × 452) mod 50306347
= 40090771 (mod 50306347)
So our answer is 40090771 (mod 50306347).
Verification
So we found that x ≡ 40090771
If this is correct, then the following statements (i.e. the original equations) are true:
238x (mod 481) ≡ 726 (mod 481)
211x (mod 223) ≡ 244 (mod 223)
580x (mod 469) ≡ 504 (mod 469)
Let's see whether that's indeed the case if we use x ≡ 40090771.