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 |
|---|---|---|---|---|---|---|
| 698 | 551 | 1 | 147 | 0 | 1 | -1 |
| 551 | 147 | 3 | 110 | 1 | -1 | 4 |
| 147 | 110 | 1 | 37 | -1 | 4 | -5 |
| 110 | 37 | 2 | 36 | 4 | -5 | 14 |
| 37 | 36 | 1 | 1 | -5 | 14 | -19 |
| 36 | 1 | 36 | 0 | 14 | -19 | 698 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 95 | 49 | 1 | 46 | 0 | 1 | -1 |
| 49 | 46 | 1 | 3 | 1 | -1 | 2 |
| 46 | 3 | 15 | 1 | -1 | 2 | -31 |
| 3 | 1 | 3 | 0 | 2 | -31 | 95 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 517 | 947 | 0 | 517 | 0 | 1 | 0 |
| 947 | 517 | 1 | 430 | 1 | 0 | 1 |
| 517 | 430 | 1 | 87 | 0 | 1 | -1 |
| 430 | 87 | 4 | 82 | 1 | -1 | 5 |
| 87 | 82 | 1 | 5 | -1 | 5 | -6 |
| 82 | 5 | 16 | 2 | 5 | -6 | 101 |
| 5 | 2 | 2 | 1 | -6 | 101 | -208 |
| 2 | 1 | 2 | 0 | 101 | -208 | 517 |
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 ≡ 961 × 551-1 (mod 698) ≡ 961 × 679 (mod 698) ≡ 587 (mod 698)x ≡ 391 × 49-1 (mod 95) ≡ 391 × 64 (mod 95) ≡ 39 (mod 95)x ≡ 995 × 947-1 (mod 517) ≡ 995 × 309 (mod 517) ≡ 357 (mod 517)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 587 (mod 698)
x ≡ 39 (mod 95)
x ≡ 357 (mod 517)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 698 × 95 × 517 = 34282270 - We calculate the numbers M1 to M3
M1=M/m1=34282270/698=49115,M2=M/m2=34282270/95=360866,M3=M/m3=34282270/517=66310 - 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 323 mod 698 ≡ 323n b q r t1 t2 t3 698 49115 0 698 0 1 0 49115 698 70 255 1 0 1 698 255 2 188 0 1 -2 255 188 1 67 1 -2 3 188 67 2 54 -2 3 -8 67 54 1 13 3 -8 11 54 13 4 2 -8 11 -52 13 2 6 1 11 -52 323 2 1 2 0 -52 323 -698
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -39 mod 95 ≡ 56n b q r t1 t2 t3 95 360866 0 95 0 1 0 360866 95 3798 56 1 0 1 95 56 1 39 0 1 -1 56 39 1 17 1 -1 2 39 17 2 5 -1 2 -5 17 5 3 2 2 -5 17 5 2 2 1 -5 17 -39 2 1 2 0 17 -39 95
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -27 mod 517 ≡ 490n b q r t1 t2 t3 517 66310 0 517 0 1 0 66310 517 128 134 1 0 1 517 134 3 115 0 1 -3 134 115 1 19 1 -3 4 115 19 6 1 -3 4 -27 19 1 19 0 4 -27 517
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
= (587 × 49115 × 323 +
39 × 360866 × 56 +
357 × 66310 × 490) mod 34282270
= 33598119 (mod 34282270)
So our answer is 33598119 (mod 34282270).
Verification
So we found that x ≡ 33598119
If this is correct, then the following statements (i.e. the original equations) are true:
551x (mod 698) ≡ 961 (mod 698)
49x (mod 95) ≡ 391 (mod 95)
947x (mod 517) ≡ 995 (mod 517)
Let's see whether that's indeed the case if we use x ≡ 33598119.