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 |
|---|---|---|---|---|---|---|
| 321 | 305 | 1 | 16 | 0 | 1 | -1 |
| 305 | 16 | 19 | 1 | 1 | -1 | 20 |
| 16 | 1 | 16 | 0 | -1 | 20 | -321 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 373 | 96 | 3 | 85 | 0 | 1 | -3 |
| 96 | 85 | 1 | 11 | 1 | -3 | 4 |
| 85 | 11 | 7 | 8 | -3 | 4 | -31 |
| 11 | 8 | 1 | 3 | 4 | -31 | 35 |
| 8 | 3 | 2 | 2 | -31 | 35 | -101 |
| 3 | 2 | 1 | 1 | 35 | -101 | 136 |
| 2 | 1 | 2 | 0 | -101 | 136 | -373 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 79 | 371 | 0 | 79 | 0 | 1 | 0 |
| 371 | 79 | 4 | 55 | 1 | 0 | 1 |
| 79 | 55 | 1 | 24 | 0 | 1 | -1 |
| 55 | 24 | 2 | 7 | 1 | -1 | 3 |
| 24 | 7 | 3 | 3 | -1 | 3 | -10 |
| 7 | 3 | 2 | 1 | 3 | -10 | 23 |
| 3 | 1 | 3 | 0 | -10 | 23 | -79 |
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 ≡ 224 × 305-1 (mod 321) ≡ 224 × 20 (mod 321) ≡ 307 (mod 321)x ≡ 5 × 96-1 (mod 373) ≡ 5 × 136 (mod 373) ≡ 307 (mod 373)x ≡ 145 × 371-1 (mod 79) ≡ 145 × 23 (mod 79) ≡ 17 (mod 79)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 307 (mod 321)
x ≡ 307 (mod 373)
x ≡ 17 (mod 79)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 321 × 373 × 79 = 9458907 - We calculate the numbers M1 to M3
M1=M/m1=9458907/321=29467,M2=M/m2=9458907/373=25359,M3=M/m3=9458907/79=119733 - 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 79 mod 321 ≡ 79n b q r t1 t2 t3 321 29467 0 321 0 1 0 29467 321 91 256 1 0 1 321 256 1 65 0 1 -1 256 65 3 61 1 -1 4 65 61 1 4 -1 4 -5 61 4 15 1 4 -5 79 4 1 4 0 -5 79 -321
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 149 mod 373 ≡ 149n b q r t1 t2 t3 373 25359 0 373 0 1 0 25359 373 67 368 1 0 1 373 368 1 5 0 1 -1 368 5 73 3 1 -1 74 5 3 1 2 -1 74 -75 3 2 1 1 74 -75 149 2 1 2 0 -75 149 -373
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 28 mod 79 ≡ 28n b q r t1 t2 t3 79 119733 0 79 0 1 0 119733 79 1515 48 1 0 1 79 48 1 31 0 1 -1 48 31 1 17 1 -1 2 31 17 1 14 -1 2 -3 17 14 1 3 2 -3 5 14 3 4 2 -3 5 -23 3 2 1 1 5 -23 28 2 1 2 0 -23 28 -79
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
= (307 × 29467 × 79 +
307 × 25359 × 149 +
17 × 119733 × 28) mod 9458907
= 2035768 (mod 9458907)
So our answer is 2035768 (mod 9458907).
Verification
So we found that x ≡ 2035768
If this is correct, then the following statements (i.e. the original equations) are true:
305x (mod 321) ≡ 224 (mod 321)
96x (mod 373) ≡ 5 (mod 373)
371x (mod 79) ≡ 145 (mod 79)
Let's see whether that's indeed the case if we use x ≡ 2035768.