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 |
|---|---|---|---|---|---|---|
| 683 | 32 | 21 | 11 | 0 | 1 | -21 |
| 32 | 11 | 2 | 10 | 1 | -21 | 43 |
| 11 | 10 | 1 | 1 | -21 | 43 | -64 |
| 10 | 1 | 10 | 0 | 43 | -64 | 683 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 479 | 703 | 0 | 479 | 0 | 1 | 0 |
| 703 | 479 | 1 | 224 | 1 | 0 | 1 |
| 479 | 224 | 2 | 31 | 0 | 1 | -2 |
| 224 | 31 | 7 | 7 | 1 | -2 | 15 |
| 31 | 7 | 4 | 3 | -2 | 15 | -62 |
| 7 | 3 | 2 | 1 | 15 | -62 | 139 |
| 3 | 1 | 3 | 0 | -62 | 139 | -479 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 549 | 14 | 39 | 3 | 0 | 1 | -39 |
| 14 | 3 | 4 | 2 | 1 | -39 | 157 |
| 3 | 2 | 1 | 1 | -39 | 157 | -196 |
| 2 | 1 | 2 | 0 | 157 | -196 | 549 |
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 ≡ 517 × 32-1 (mod 683) ≡ 517 × 619 (mod 683) ≡ 379 (mod 683)x ≡ 379 × 703-1 (mod 479) ≡ 379 × 139 (mod 479) ≡ 470 (mod 479)x ≡ 872 × 14-1 (mod 549) ≡ 872 × 353 (mod 549) ≡ 376 (mod 549)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 379 (mod 683)
x ≡ 470 (mod 479)
x ≡ 376 (mod 549)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 683 × 479 × 549 = 179609193 - We calculate the numbers M1 to M3
M1=M/m1=179609193/683=262971,M2=M/m2=179609193/479=374967,M3=M/m3=179609193/549=327157 - 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 -128 mod 683 ≡ 555n b q r t1 t2 t3 683 262971 0 683 0 1 0 262971 683 385 16 1 0 1 683 16 42 11 0 1 -42 16 11 1 5 1 -42 43 11 5 2 1 -42 43 -128 5 1 5 0 43 -128 683
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -165 mod 479 ≡ 314n b q r t1 t2 t3 479 374967 0 479 0 1 0 374967 479 782 389 1 0 1 479 389 1 90 0 1 -1 389 90 4 29 1 -1 5 90 29 3 3 -1 5 -16 29 3 9 2 5 -16 149 3 2 1 1 -16 149 -165 2 1 2 0 149 -165 479
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -257 mod 549 ≡ 292n b q r t1 t2 t3 549 327157 0 549 0 1 0 327157 549 595 502 1 0 1 549 502 1 47 0 1 -1 502 47 10 32 1 -1 11 47 32 1 15 -1 11 -12 32 15 2 2 11 -12 35 15 2 7 1 -12 35 -257 2 1 2 0 35 -257 549
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
= (379 × 262971 × 555 +
470 × 374967 × 314 +
376 × 327157 × 292) mod 179609193
= 10384711 (mod 179609193)
So our answer is 10384711 (mod 179609193).
Verification
So we found that x ≡ 10384711
If this is correct, then the following statements (i.e. the original equations) are true:
32x (mod 683) ≡ 517 (mod 683)
703x (mod 479) ≡ 379 (mod 479)
14x (mod 549) ≡ 872 (mod 549)
Let's see whether that's indeed the case if we use x ≡ 10384711.