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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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 |
|---|---|---|---|---|---|---|
| 247 | 250 | 0 | 247 | 0 | 1 | 0 |
| 250 | 247 | 1 | 3 | 1 | 0 | 1 |
| 247 | 3 | 82 | 1 | 0 | 1 | -82 |
| 3 | 1 | 3 | 0 | 1 | -82 | 247 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 769 | 627 | 1 | 142 | 0 | 1 | -1 |
| 627 | 142 | 4 | 59 | 1 | -1 | 5 |
| 142 | 59 | 2 | 24 | -1 | 5 | -11 |
| 59 | 24 | 2 | 11 | 5 | -11 | 27 |
| 24 | 11 | 2 | 2 | -11 | 27 | -65 |
| 11 | 2 | 5 | 1 | 27 | -65 | 352 |
| 2 | 1 | 2 | 0 | -65 | 352 | -769 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 827 | 895 | 0 | 827 | 0 | 1 | 0 |
| 895 | 827 | 1 | 68 | 1 | 0 | 1 |
| 827 | 68 | 12 | 11 | 0 | 1 | -12 |
| 68 | 11 | 6 | 2 | 1 | -12 | 73 |
| 11 | 2 | 5 | 1 | -12 | 73 | -377 |
| 2 | 1 | 2 | 0 | 73 | -377 | 827 |
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 ≡ 735 × 250-1 (mod 247) ≡ 735 × 165 (mod 247) ≡ 245 (mod 247)x ≡ 139 × 627-1 (mod 769) ≡ 139 × 352 (mod 769) ≡ 481 (mod 769)x ≡ 962 × 895-1 (mod 827) ≡ 962 × 450 (mod 827) ≡ 379 (mod 827)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 245 (mod 247)
x ≡ 481 (mod 769)
x ≡ 379 (mod 827)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 247 × 769 × 827 = 157082861 - We calculate the numbers M1 to M3
M1=M/m1=157082861/247=635963,M2=M/m2=157082861/769=204269,M3=M/m3=157082861/827=189943 - 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 -4 mod 247 ≡ 243n b q r t1 t2 t3 247 635963 0 247 0 1 0 635963 247 2574 185 1 0 1 247 185 1 62 0 1 -1 185 62 2 61 1 -1 3 62 61 1 1 -1 3 -4 61 1 61 0 3 -4 247
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -313 mod 769 ≡ 456n b q r t1 t2 t3 769 204269 0 769 0 1 0 204269 769 265 484 1 0 1 769 484 1 285 0 1 -1 484 285 1 199 1 -1 2 285 199 1 86 -1 2 -3 199 86 2 27 2 -3 8 86 27 3 5 -3 8 -27 27 5 5 2 8 -27 143 5 2 2 1 -27 143 -313 2 1 2 0 143 -313 769
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 350 mod 827 ≡ 350n b q r t1 t2 t3 827 189943 0 827 0 1 0 189943 827 229 560 1 0 1 827 560 1 267 0 1 -1 560 267 2 26 1 -1 3 267 26 10 7 -1 3 -31 26 7 3 5 3 -31 96 7 5 1 2 -31 96 -127 5 2 2 1 96 -127 350 2 1 2 0 -127 350 -827
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
= (245 × 635963 × 243 +
481 × 204269 × 456 +
379 × 189943 × 350) mod 157082861
= 102698893 (mod 157082861)
So our answer is 102698893 (mod 157082861).
Verification
So we found that x ≡ 102698893
If this is correct, then the following statements (i.e. the original equations) are true:
250x (mod 247) ≡ 735 (mod 247)
627x (mod 769) ≡ 139 (mod 769)
895x (mod 827) ≡ 962 (mod 827)
Let's see whether that's indeed the case if we use x ≡ 102698893.