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.
This fills all textboxes with random numbers. If you fill in random numbers yourself, it is very likely that those numbers do not have a solution. To avoid disappointment, use this button instead! It only uses random numbers that do have a solution.
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 |
|---|---|---|---|---|---|---|
| 839 | 54 | 15 | 29 | 0 | 1 | -15 |
| 54 | 29 | 1 | 25 | 1 | -15 | 16 |
| 29 | 25 | 1 | 4 | -15 | 16 | -31 |
| 25 | 4 | 6 | 1 | 16 | -31 | 202 |
| 4 | 1 | 4 | 0 | -31 | 202 | -839 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 757 | 651 | 1 | 106 | 0 | 1 | -1 |
| 651 | 106 | 6 | 15 | 1 | -1 | 7 |
| 106 | 15 | 7 | 1 | -1 | 7 | -50 |
| 15 | 1 | 15 | 0 | 7 | -50 | 757 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 907 | 323 | 2 | 261 | 0 | 1 | -2 |
| 323 | 261 | 1 | 62 | 1 | -2 | 3 |
| 261 | 62 | 4 | 13 | -2 | 3 | -14 |
| 62 | 13 | 4 | 10 | 3 | -14 | 59 |
| 13 | 10 | 1 | 3 | -14 | 59 | -73 |
| 10 | 3 | 3 | 1 | 59 | -73 | 278 |
| 3 | 1 | 3 | 0 | -73 | 278 | -907 |
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 ≡ 661 × 54-1 (mod 839) ≡ 661 × 202 (mod 839) ≡ 121 (mod 839)x ≡ 762 × 651-1 (mod 757) ≡ 762 × 707 (mod 757) ≡ 507 (mod 757)x ≡ 235 × 323-1 (mod 907) ≡ 235 × 278 (mod 907) ≡ 26 (mod 907)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 121 (mod 839)
x ≡ 507 (mod 757)
x ≡ 26 (mod 907)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 839 × 757 × 907 = 576056561 - We calculate the numbers M1 to M3
M1=M/m1=576056561/839=686599,M2=M/m2=576056561/757=760973,M3=M/m3=576056561/907=635123 - 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 113 mod 839 ≡ 113n b q r t1 t2 t3 839 686599 0 839 0 1 0 686599 839 818 297 1 0 1 839 297 2 245 0 1 -2 297 245 1 52 1 -2 3 245 52 4 37 -2 3 -14 52 37 1 15 3 -14 17 37 15 2 7 -14 17 -48 15 7 2 1 17 -48 113 7 1 7 0 -48 113 -839
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 302 mod 757 ≡ 302n b q r t1 t2 t3 757 760973 0 757 0 1 0 760973 757 1005 188 1 0 1 757 188 4 5 0 1 -4 188 5 37 3 1 -4 149 5 3 1 2 -4 149 -153 3 2 1 1 149 -153 302 2 1 2 0 -153 302 -757
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 423 mod 907 ≡ 423n b q r t1 t2 t3 907 635123 0 907 0 1 0 635123 907 700 223 1 0 1 907 223 4 15 0 1 -4 223 15 14 13 1 -4 57 15 13 1 2 -4 57 -61 13 2 6 1 57 -61 423 2 1 2 0 -61 423 -907
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
= (121 × 686599 × 113 +
507 × 760973 × 302 +
26 × 635123 × 423) mod 576056561
= 395561773 (mod 576056561)
So our answer is 395561773 (mod 576056561).
Verification
So we found that x ≡ 395561773
If this is correct, then the following statements (i.e. the original equations) are true:
54x (mod 839) ≡ 661 (mod 839)
651x (mod 757) ≡ 762 (mod 757)
323x (mod 907) ≡ 235 (mod 907)
Let's see whether that's indeed the case if we use x ≡ 395561773.