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 |
|---|---|---|---|---|---|---|
| 353 | 981 | 0 | 353 | 0 | 1 | 0 |
| 981 | 353 | 2 | 275 | 1 | 0 | 1 |
| 353 | 275 | 1 | 78 | 0 | 1 | -1 |
| 275 | 78 | 3 | 41 | 1 | -1 | 4 |
| 78 | 41 | 1 | 37 | -1 | 4 | -5 |
| 41 | 37 | 1 | 4 | 4 | -5 | 9 |
| 37 | 4 | 9 | 1 | -5 | 9 | -86 |
| 4 | 1 | 4 | 0 | 9 | -86 | 353 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 421 | 796 | 0 | 421 | 0 | 1 | 0 |
| 796 | 421 | 1 | 375 | 1 | 0 | 1 |
| 421 | 375 | 1 | 46 | 0 | 1 | -1 |
| 375 | 46 | 8 | 7 | 1 | -1 | 9 |
| 46 | 7 | 6 | 4 | -1 | 9 | -55 |
| 7 | 4 | 1 | 3 | 9 | -55 | 64 |
| 4 | 3 | 1 | 1 | -55 | 64 | -119 |
| 3 | 1 | 3 | 0 | 64 | -119 | 421 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 373 | 547 | 0 | 373 | 0 | 1 | 0 |
| 547 | 373 | 1 | 174 | 1 | 0 | 1 |
| 373 | 174 | 2 | 25 | 0 | 1 | -2 |
| 174 | 25 | 6 | 24 | 1 | -2 | 13 |
| 25 | 24 | 1 | 1 | -2 | 13 | -15 |
| 24 | 1 | 24 | 0 | 13 | -15 | 373 |
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 ≡ 140 × 981-1 (mod 353) ≡ 140 × 267 (mod 353) ≡ 315 (mod 353)x ≡ 848 × 796-1 (mod 421) ≡ 848 × 302 (mod 421) ≡ 128 (mod 421)x ≡ 297 × 547-1 (mod 373) ≡ 297 × 358 (mod 373) ≡ 21 (mod 373)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 315 (mod 353)
x ≡ 128 (mod 421)
x ≡ 21 (mod 373)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 353 × 421 × 373 = 55432649 - We calculate the numbers M1 to M3
M1=M/m1=55432649/353=157033,M2=M/m2=55432649/421=131669,M3=M/m3=55432649/373=148613 - 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 -129 mod 353 ≡ 224n b q r t1 t2 t3 353 157033 0 353 0 1 0 157033 353 444 301 1 0 1 353 301 1 52 0 1 -1 301 52 5 41 1 -1 6 52 41 1 11 -1 6 -7 41 11 3 8 6 -7 27 11 8 1 3 -7 27 -34 8 3 2 2 27 -34 95 3 2 1 1 -34 95 -129 2 1 2 0 95 -129 353
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 85 mod 421 ≡ 85n b q r t1 t2 t3 421 131669 0 421 0 1 0 131669 421 312 317 1 0 1 421 317 1 104 0 1 -1 317 104 3 5 1 -1 4 104 5 20 4 -1 4 -81 5 4 1 1 4 -81 85 4 1 4 0 -81 85 -421
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 61 mod 373 ≡ 61n b q r t1 t2 t3 373 148613 0 373 0 1 0 148613 373 398 159 1 0 1 373 159 2 55 0 1 -2 159 55 2 49 1 -2 5 55 49 1 6 -2 5 -7 49 6 8 1 5 -7 61 6 1 6 0 -7 61 -373
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
= (315 × 157033 × 224 +
128 × 131669 × 85 +
21 × 148613 × 61) mod 55432649
= 9103832 (mod 55432649)
So our answer is 9103832 (mod 55432649).
Verification
So we found that x ≡ 9103832
If this is correct, then the following statements (i.e. the original equations) are true:
981x (mod 353) ≡ 140 (mod 353)
796x (mod 421) ≡ 848 (mod 421)
547x (mod 373) ≡ 297 (mod 373)
Let's see whether that's indeed the case if we use x ≡ 9103832.