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 |
|---|---|---|---|---|---|---|
| 487 | 913 | 0 | 487 | 0 | 1 | 0 |
| 913 | 487 | 1 | 426 | 1 | 0 | 1 |
| 487 | 426 | 1 | 61 | 0 | 1 | -1 |
| 426 | 61 | 6 | 60 | 1 | -1 | 7 |
| 61 | 60 | 1 | 1 | -1 | 7 | -8 |
| 60 | 1 | 60 | 0 | 7 | -8 | 487 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 671 | 593 | 1 | 78 | 0 | 1 | -1 |
| 593 | 78 | 7 | 47 | 1 | -1 | 8 |
| 78 | 47 | 1 | 31 | -1 | 8 | -9 |
| 47 | 31 | 1 | 16 | 8 | -9 | 17 |
| 31 | 16 | 1 | 15 | -9 | 17 | -26 |
| 16 | 15 | 1 | 1 | 17 | -26 | 43 |
| 15 | 1 | 15 | 0 | -26 | 43 | -671 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 934 | 509 | 1 | 425 | 0 | 1 | -1 |
| 509 | 425 | 1 | 84 | 1 | -1 | 2 |
| 425 | 84 | 5 | 5 | -1 | 2 | -11 |
| 84 | 5 | 16 | 4 | 2 | -11 | 178 |
| 5 | 4 | 1 | 1 | -11 | 178 | -189 |
| 4 | 1 | 4 | 0 | 178 | -189 | 934 |
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 ≡ 947 × 913-1 (mod 487) ≡ 947 × 479 (mod 487) ≡ 216 (mod 487)x ≡ 980 × 593-1 (mod 671) ≡ 980 × 43 (mod 671) ≡ 538 (mod 671)x ≡ 549 × 509-1 (mod 934) ≡ 549 × 745 (mod 934) ≡ 847 (mod 934)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 216 (mod 487)
x ≡ 538 (mod 671)
x ≡ 847 (mod 934)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 487 × 671 × 934 = 305209718 - We calculate the numbers M1 to M3
M1=M/m1=305209718/487=626714,M2=M/m2=305209718/671=454858,M3=M/m3=305209718/934=326777 - 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 -62 mod 487 ≡ 425n b q r t1 t2 t3 487 626714 0 487 0 1 0 626714 487 1286 432 1 0 1 487 432 1 55 0 1 -1 432 55 7 47 1 -1 8 55 47 1 8 -1 8 -9 47 8 5 7 8 -9 53 8 7 1 1 -9 53 -62 7 1 7 0 53 -62 487
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 260 mod 671 ≡ 260n b q r t1 t2 t3 671 454858 0 671 0 1 0 454858 671 677 591 1 0 1 671 591 1 80 0 1 -1 591 80 7 31 1 -1 8 80 31 2 18 -1 8 -17 31 18 1 13 8 -17 25 18 13 1 5 -17 25 -42 13 5 2 3 25 -42 109 5 3 1 2 -42 109 -151 3 2 1 1 109 -151 260 2 1 2 0 -151 260 -671
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -243 mod 934 ≡ 691n b q r t1 t2 t3 934 326777 0 934 0 1 0 326777 934 349 811 1 0 1 934 811 1 123 0 1 -1 811 123 6 73 1 -1 7 123 73 1 50 -1 7 -8 73 50 1 23 7 -8 15 50 23 2 4 -8 15 -38 23 4 5 3 15 -38 205 4 3 1 1 -38 205 -243 3 1 3 0 205 -243 934
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
= (216 × 626714 × 425 +
538 × 454858 × 260 +
847 × 326777 × 691) mod 305209718
= 183402955 (mod 305209718)
So our answer is 183402955 (mod 305209718).
Verification
So we found that x ≡ 183402955
If this is correct, then the following statements (i.e. the original equations) are true:
913x (mod 487) ≡ 947 (mod 487)
593x (mod 671) ≡ 980 (mod 671)
509x (mod 934) ≡ 549 (mod 934)
Let's see whether that's indeed the case if we use x ≡ 183402955.