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 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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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 |
|---|---|---|---|---|---|---|
| 961 | 22 | 43 | 15 | 0 | 1 | -43 |
| 22 | 15 | 1 | 7 | 1 | -43 | 44 |
| 15 | 7 | 2 | 1 | -43 | 44 | -131 |
| 7 | 1 | 7 | 0 | 44 | -131 | 961 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 47 | 93 | 0 | 47 | 0 | 1 | 0 |
| 93 | 47 | 1 | 46 | 1 | 0 | 1 |
| 47 | 46 | 1 | 1 | 0 | 1 | -1 |
| 46 | 1 | 46 | 0 | 1 | -1 | 47 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 653 | 452 | 1 | 201 | 0 | 1 | -1 |
| 452 | 201 | 2 | 50 | 1 | -1 | 3 |
| 201 | 50 | 4 | 1 | -1 | 3 | -13 |
| 50 | 1 | 50 | 0 | 3 | -13 | 653 |
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 ≡ 567 × 22-1 (mod 961) ≡ 567 × 830 (mod 961) ≡ 681 (mod 961)x ≡ 897 × 93-1 (mod 47) ≡ 897 × 46 (mod 47) ≡ 43 (mod 47)x ≡ 420 × 452-1 (mod 653) ≡ 420 × 640 (mod 653) ≡ 417 (mod 653)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 681 (mod 961)
x ≡ 43 (mod 47)
x ≡ 417 (mod 653)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 961 × 47 × 653 = 29494051 - We calculate the numbers M1 to M3
M1=M/m1=29494051/961=30691,M2=M/m2=29494051/47=627533,M3=M/m3=29494051/653=45167 - 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 63 mod 961 ≡ 63n b q r t1 t2 t3 961 30691 0 961 0 1 0 30691 961 31 900 1 0 1 961 900 1 61 0 1 -1 900 61 14 46 1 -1 15 61 46 1 15 -1 15 -16 46 15 3 1 15 -16 63 15 1 15 0 -16 63 -961
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 17 mod 47 ≡ 17n b q r t1 t2 t3 47 627533 0 47 0 1 0 627533 47 13351 36 1 0 1 47 36 1 11 0 1 -1 36 11 3 3 1 -1 4 11 3 3 2 -1 4 -13 3 2 1 1 4 -13 17 2 1 2 0 -13 17 -47
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -279 mod 653 ≡ 374n b q r t1 t2 t3 653 45167 0 653 0 1 0 45167 653 69 110 1 0 1 653 110 5 103 0 1 -5 110 103 1 7 1 -5 6 103 7 14 5 -5 6 -89 7 5 1 2 6 -89 95 5 2 2 1 -89 95 -279 2 1 2 0 95 -279 653
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
= (681 × 30691 × 63 +
43 × 627533 × 17 +
417 × 45167 × 374) mod 29494051
= 896333 (mod 29494051)
So our answer is 896333 (mod 29494051).
Verification
So we found that x ≡ 896333
If this is correct, then the following statements (i.e. the original equations) are true:
22x (mod 961) ≡ 567 (mod 961)
93x (mod 47) ≡ 897 (mod 47)
452x (mod 653) ≡ 420 (mod 653)
Let's see whether that's indeed the case if we use x ≡ 896333.