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 |
|---|---|---|---|---|---|---|
| 631 | 688 | 0 | 631 | 0 | 1 | 0 |
| 688 | 631 | 1 | 57 | 1 | 0 | 1 |
| 631 | 57 | 11 | 4 | 0 | 1 | -11 |
| 57 | 4 | 14 | 1 | 1 | -11 | 155 |
| 4 | 1 | 4 | 0 | -11 | 155 | -631 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 977 | 229 | 4 | 61 | 0 | 1 | -4 |
| 229 | 61 | 3 | 46 | 1 | -4 | 13 |
| 61 | 46 | 1 | 15 | -4 | 13 | -17 |
| 46 | 15 | 3 | 1 | 13 | -17 | 64 |
| 15 | 1 | 15 | 0 | -17 | 64 | -977 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 269 | 532 | 0 | 269 | 0 | 1 | 0 |
| 532 | 269 | 1 | 263 | 1 | 0 | 1 |
| 269 | 263 | 1 | 6 | 0 | 1 | -1 |
| 263 | 6 | 43 | 5 | 1 | -1 | 44 |
| 6 | 5 | 1 | 1 | -1 | 44 | -45 |
| 5 | 1 | 5 | 0 | 44 | -45 | 269 |
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 ≡ 89 × 688-1 (mod 631) ≡ 89 × 155 (mod 631) ≡ 544 (mod 631)x ≡ 553 × 229-1 (mod 977) ≡ 553 × 64 (mod 977) ≡ 220 (mod 977)x ≡ 628 × 532-1 (mod 269) ≡ 628 × 224 (mod 269) ≡ 254 (mod 269)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 544 (mod 631)
x ≡ 220 (mod 977)
x ≡ 254 (mod 269)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 631 × 977 × 269 = 165835003 - We calculate the numbers M1 to M3
M1=M/m1=165835003/631=262813,M2=M/m2=165835003/977=169739,M3=M/m3=165835003/269=616487 - 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 211 mod 631 ≡ 211n b q r t1 t2 t3 631 262813 0 631 0 1 0 262813 631 416 317 1 0 1 631 317 1 314 0 1 -1 317 314 1 3 1 -1 2 314 3 104 2 -1 2 -209 3 2 1 1 2 -209 211 2 1 2 0 -209 211 -631
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 298 mod 977 ≡ 298n b q r t1 t2 t3 977 169739 0 977 0 1 0 169739 977 173 718 1 0 1 977 718 1 259 0 1 -1 718 259 2 200 1 -1 3 259 200 1 59 -1 3 -4 200 59 3 23 3 -4 15 59 23 2 13 -4 15 -34 23 13 1 10 15 -34 49 13 10 1 3 -34 49 -83 10 3 3 1 49 -83 298 3 1 3 0 -83 298 -977
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 97 mod 269 ≡ 97n b q r t1 t2 t3 269 616487 0 269 0 1 0 616487 269 2291 208 1 0 1 269 208 1 61 0 1 -1 208 61 3 25 1 -1 4 61 25 2 11 -1 4 -9 25 11 2 3 4 -9 22 11 3 3 2 -9 22 -75 3 2 1 1 22 -75 97 2 1 2 0 -75 97 -269
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
= (544 × 262813 × 211 +
220 × 169739 × 298 +
254 × 616487 × 97) mod 165835003
= 99921918 (mod 165835003)
So our answer is 99921918 (mod 165835003).
Verification
So we found that x ≡ 99921918
If this is correct, then the following statements (i.e. the original equations) are true:
688x (mod 631) ≡ 89 (mod 631)
229x (mod 977) ≡ 553 (mod 977)
532x (mod 269) ≡ 628 (mod 269)
Let's see whether that's indeed the case if we use x ≡ 99921918.