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 |
|---|---|---|---|---|---|---|
| 691 | 83 | 8 | 27 | 0 | 1 | -8 |
| 83 | 27 | 3 | 2 | 1 | -8 | 25 |
| 27 | 2 | 13 | 1 | -8 | 25 | -333 |
| 2 | 1 | 2 | 0 | 25 | -333 | 691 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 892 | 891 | 1 | 1 | 0 | 1 | -1 |
| 891 | 1 | 891 | 0 | 1 | -1 | 892 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 787 | 861 | 0 | 787 | 0 | 1 | 0 |
| 861 | 787 | 1 | 74 | 1 | 0 | 1 |
| 787 | 74 | 10 | 47 | 0 | 1 | -10 |
| 74 | 47 | 1 | 27 | 1 | -10 | 11 |
| 47 | 27 | 1 | 20 | -10 | 11 | -21 |
| 27 | 20 | 1 | 7 | 11 | -21 | 32 |
| 20 | 7 | 2 | 6 | -21 | 32 | -85 |
| 7 | 6 | 1 | 1 | 32 | -85 | 117 |
| 6 | 1 | 6 | 0 | -85 | 117 | -787 |
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 ≡ 465 × 83-1 (mod 691) ≡ 465 × 358 (mod 691) ≡ 630 (mod 691)x ≡ 245 × 891-1 (mod 892) ≡ 245 × 891 (mod 892) ≡ 647 (mod 892)x ≡ 863 × 861-1 (mod 787) ≡ 863 × 117 (mod 787) ≡ 235 (mod 787)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 630 (mod 691)
x ≡ 647 (mod 892)
x ≡ 235 (mod 787)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 691 × 892 × 787 = 485084764 - We calculate the numbers M1 to M3
M1=M/m1=485084764/691=702004,M2=M/m2=485084764/892=543817,M3=M/m3=485084764/787=616372 - 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 93 mod 691 ≡ 93n b q r t1 t2 t3 691 702004 0 691 0 1 0 702004 691 1015 639 1 0 1 691 639 1 52 0 1 -1 639 52 12 15 1 -1 13 52 15 3 7 -1 13 -40 15 7 2 1 13 -40 93 7 1 7 0 -40 93 -691
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -315 mod 892 ≡ 577n b q r t1 t2 t3 892 543817 0 892 0 1 0 543817 892 609 589 1 0 1 892 589 1 303 0 1 -1 589 303 1 286 1 -1 2 303 286 1 17 -1 2 -3 286 17 16 14 2 -3 50 17 14 1 3 -3 50 -53 14 3 4 2 50 -53 262 3 2 1 1 -53 262 -315 2 1 2 0 262 -315 892
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 172 mod 787 ≡ 172n b q r t1 t2 t3 787 616372 0 787 0 1 0 616372 787 783 151 1 0 1 787 151 5 32 0 1 -5 151 32 4 23 1 -5 21 32 23 1 9 -5 21 -26 23 9 2 5 21 -26 73 9 5 1 4 -26 73 -99 5 4 1 1 73 -99 172 4 1 4 0 -99 172 -787
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
= (630 × 702004 × 93 +
647 × 543817 × 577 +
235 × 616372 × 172) mod 485084764
= 324429967 (mod 485084764)
So our answer is 324429967 (mod 485084764).
Verification
So we found that x ≡ 324429967
If this is correct, then the following statements (i.e. the original equations) are true:
83x (mod 691) ≡ 465 (mod 691)
891x (mod 892) ≡ 245 (mod 892)
861x (mod 787) ≡ 863 (mod 787)
Let's see whether that's indeed the case if we use x ≡ 324429967.