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" .
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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 |
|---|---|---|---|---|---|---|
| 491 | 556 | 0 | 491 | 0 | 1 | 0 |
| 556 | 491 | 1 | 65 | 1 | 0 | 1 |
| 491 | 65 | 7 | 36 | 0 | 1 | -7 |
| 65 | 36 | 1 | 29 | 1 | -7 | 8 |
| 36 | 29 | 1 | 7 | -7 | 8 | -15 |
| 29 | 7 | 4 | 1 | 8 | -15 | 68 |
| 7 | 1 | 7 | 0 | -15 | 68 | -491 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 463 | 201 | 2 | 61 | 0 | 1 | -2 |
| 201 | 61 | 3 | 18 | 1 | -2 | 7 |
| 61 | 18 | 3 | 7 | -2 | 7 | -23 |
| 18 | 7 | 2 | 4 | 7 | -23 | 53 |
| 7 | 4 | 1 | 3 | -23 | 53 | -76 |
| 4 | 3 | 1 | 1 | 53 | -76 | 129 |
| 3 | 1 | 3 | 0 | -76 | 129 | -463 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 569 | 101 | 5 | 64 | 0 | 1 | -5 |
| 101 | 64 | 1 | 37 | 1 | -5 | 6 |
| 64 | 37 | 1 | 27 | -5 | 6 | -11 |
| 37 | 27 | 1 | 10 | 6 | -11 | 17 |
| 27 | 10 | 2 | 7 | -11 | 17 | -45 |
| 10 | 7 | 1 | 3 | 17 | -45 | 62 |
| 7 | 3 | 2 | 1 | -45 | 62 | -169 |
| 3 | 1 | 3 | 0 | 62 | -169 | 569 |
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 ≡ 334 × 556-1 (mod 491) ≡ 334 × 68 (mod 491) ≡ 126 (mod 491)x ≡ 511 × 201-1 (mod 463) ≡ 511 × 129 (mod 463) ≡ 173 (mod 463)x ≡ 957 × 101-1 (mod 569) ≡ 957 × 400 (mod 569) ≡ 432 (mod 569)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 126 (mod 491)
x ≡ 173 (mod 463)
x ≡ 432 (mod 569)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 491 × 463 × 569 = 129352477 - We calculate the numbers M1 to M3
M1=M/m1=129352477/491=263447,M2=M/m2=129352477/463=279379,M3=M/m3=129352477/569=227333 - 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 -154 mod 491 ≡ 337n b q r t1 t2 t3 491 263447 0 491 0 1 0 263447 491 536 271 1 0 1 491 271 1 220 0 1 -1 271 220 1 51 1 -1 2 220 51 4 16 -1 2 -9 51 16 3 3 2 -9 29 16 3 5 1 -9 29 -154 3 1 3 0 29 -154 491
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -212 mod 463 ≡ 251n b q r t1 t2 t3 463 279379 0 463 0 1 0 279379 463 603 190 1 0 1 463 190 2 83 0 1 -2 190 83 2 24 1 -2 5 83 24 3 11 -2 5 -17 24 11 2 2 5 -17 39 11 2 5 1 -17 39 -212 2 1 2 0 39 -212 463
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -130 mod 569 ≡ 439n b q r t1 t2 t3 569 227333 0 569 0 1 0 227333 569 399 302 1 0 1 569 302 1 267 0 1 -1 302 267 1 35 1 -1 2 267 35 7 22 -1 2 -15 35 22 1 13 2 -15 17 22 13 1 9 -15 17 -32 13 9 1 4 17 -32 49 9 4 2 1 -32 49 -130 4 1 4 0 49 -130 569
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
= (126 × 263447 × 337 +
173 × 279379 × 251 +
432 × 227333 × 439) mod 129352477
= 73388914 (mod 129352477)
So our answer is 73388914 (mod 129352477).
Verification
So we found that x ≡ 73388914
If this is correct, then the following statements (i.e. the original equations) are true:
556x (mod 491) ≡ 334 (mod 491)
201x (mod 463) ≡ 511 (mod 463)
101x (mod 569) ≡ 957 (mod 569)
Let's see whether that's indeed the case if we use x ≡ 73388914.