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 |
|---|---|---|---|---|---|---|
| 709 | 567 | 1 | 142 | 0 | 1 | -1 |
| 567 | 142 | 3 | 141 | 1 | -1 | 4 |
| 142 | 141 | 1 | 1 | -1 | 4 | -5 |
| 141 | 1 | 141 | 0 | 4 | -5 | 709 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 521 | 905 | 0 | 521 | 0 | 1 | 0 |
| 905 | 521 | 1 | 384 | 1 | 0 | 1 |
| 521 | 384 | 1 | 137 | 0 | 1 | -1 |
| 384 | 137 | 2 | 110 | 1 | -1 | 3 |
| 137 | 110 | 1 | 27 | -1 | 3 | -4 |
| 110 | 27 | 4 | 2 | 3 | -4 | 19 |
| 27 | 2 | 13 | 1 | -4 | 19 | -251 |
| 2 | 1 | 2 | 0 | 19 | -251 | 521 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 268 | 167 | 1 | 101 | 0 | 1 | -1 |
| 167 | 101 | 1 | 66 | 1 | -1 | 2 |
| 101 | 66 | 1 | 35 | -1 | 2 | -3 |
| 66 | 35 | 1 | 31 | 2 | -3 | 5 |
| 35 | 31 | 1 | 4 | -3 | 5 | -8 |
| 31 | 4 | 7 | 3 | 5 | -8 | 61 |
| 4 | 3 | 1 | 1 | -8 | 61 | -69 |
| 3 | 1 | 3 | 0 | 61 | -69 | 268 |
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 ≡ 406 × 567-1 (mod 709) ≡ 406 × 704 (mod 709) ≡ 97 (mod 709)x ≡ 577 × 905-1 (mod 521) ≡ 577 × 270 (mod 521) ≡ 11 (mod 521)x ≡ 239 × 167-1 (mod 268) ≡ 239 × 199 (mod 268) ≡ 125 (mod 268)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 97 (mod 709)
x ≡ 11 (mod 521)
x ≡ 125 (mod 268)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 709 × 521 × 268 = 98996252 - We calculate the numbers M1 to M3
M1=M/m1=98996252/709=139628,M2=M/m2=98996252/521=190012,M3=M/m3=98996252/268=369389 - 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 63 mod 709 ≡ 63n b q r t1 t2 t3 709 139628 0 709 0 1 0 139628 709 196 664 1 0 1 709 664 1 45 0 1 -1 664 45 14 34 1 -1 15 45 34 1 11 -1 15 -16 34 11 3 1 15 -16 63 11 1 11 0 -16 63 -709
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -126 mod 521 ≡ 395n b q r t1 t2 t3 521 190012 0 521 0 1 0 190012 521 364 368 1 0 1 521 368 1 153 0 1 -1 368 153 2 62 1 -1 3 153 62 2 29 -1 3 -7 62 29 2 4 3 -7 17 29 4 7 1 -7 17 -126 4 1 4 0 17 -126 521
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 41 mod 268 ≡ 41n b q r t1 t2 t3 268 369389 0 268 0 1 0 369389 268 1378 85 1 0 1 268 85 3 13 0 1 -3 85 13 6 7 1 -3 19 13 7 1 6 -3 19 -22 7 6 1 1 19 -22 41 6 1 6 0 -22 41 -268
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
= (97 × 139628 × 63 +
11 × 190012 × 395 +
125 × 369389 × 41) mod 98996252
= 8122401 (mod 98996252)
So our answer is 8122401 (mod 98996252).
Verification
So we found that x ≡ 8122401
If this is correct, then the following statements (i.e. the original equations) are true:
567x (mod 709) ≡ 406 (mod 709)
905x (mod 521) ≡ 577 (mod 521)
167x (mod 268) ≡ 239 (mod 268)
Let's see whether that's indeed the case if we use x ≡ 8122401.