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 |
|---|---|---|---|---|---|---|
| 277 | 755 | 0 | 277 | 0 | 1 | 0 |
| 755 | 277 | 2 | 201 | 1 | 0 | 1 |
| 277 | 201 | 1 | 76 | 0 | 1 | -1 |
| 201 | 76 | 2 | 49 | 1 | -1 | 3 |
| 76 | 49 | 1 | 27 | -1 | 3 | -4 |
| 49 | 27 | 1 | 22 | 3 | -4 | 7 |
| 27 | 22 | 1 | 5 | -4 | 7 | -11 |
| 22 | 5 | 4 | 2 | 7 | -11 | 51 |
| 5 | 2 | 2 | 1 | -11 | 51 | -113 |
| 2 | 1 | 2 | 0 | 51 | -113 | 277 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 193 | 246 | 0 | 193 | 0 | 1 | 0 |
| 246 | 193 | 1 | 53 | 1 | 0 | 1 |
| 193 | 53 | 3 | 34 | 0 | 1 | -3 |
| 53 | 34 | 1 | 19 | 1 | -3 | 4 |
| 34 | 19 | 1 | 15 | -3 | 4 | -7 |
| 19 | 15 | 1 | 4 | 4 | -7 | 11 |
| 15 | 4 | 3 | 3 | -7 | 11 | -40 |
| 4 | 3 | 1 | 1 | 11 | -40 | 51 |
| 3 | 1 | 3 | 0 | -40 | 51 | -193 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 977 | 86 | 11 | 31 | 0 | 1 | -11 |
| 86 | 31 | 2 | 24 | 1 | -11 | 23 |
| 31 | 24 | 1 | 7 | -11 | 23 | -34 |
| 24 | 7 | 3 | 3 | 23 | -34 | 125 |
| 7 | 3 | 2 | 1 | -34 | 125 | -284 |
| 3 | 1 | 3 | 0 | 125 | -284 | 977 |
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 ≡ 816 × 755-1 (mod 277) ≡ 816 × 164 (mod 277) ≡ 33 (mod 277)x ≡ 401 × 246-1 (mod 193) ≡ 401 × 51 (mod 193) ≡ 186 (mod 193)x ≡ 417 × 86-1 (mod 977) ≡ 417 × 693 (mod 977) ≡ 766 (mod 977)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 33 (mod 277)
x ≡ 186 (mod 193)
x ≡ 766 (mod 977)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 277 × 193 × 977 = 52231397 - We calculate the numbers M1 to M3
M1=M/m1=52231397/277=188561,M2=M/m2=52231397/193=270629,M3=M/m3=52231397/977=53461 - 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 277 ≡ 164n b q r t1 t2 t3 277 188561 0 277 0 1 0 188561 277 680 201 1 0 1 277 201 1 76 0 1 -1 201 76 2 49 1 -1 3 76 49 1 27 -1 3 -4 49 27 1 22 3 -4 7 27 22 1 5 -4 7 -11 22 5 4 2 7 -11 51 5 2 2 1 -11 51 -113 2 1 2 0 51 -113 277
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 9 mod 193 ≡ 9n b q r t1 t2 t3 193 270629 0 193 0 1 0 270629 193 1402 43 1 0 1 193 43 4 21 0 1 -4 43 21 2 1 1 -4 9 21 1 21 0 -4 9 -193
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 353 mod 977 ≡ 353n b q r t1 t2 t3 977 53461 0 977 0 1 0 53461 977 54 703 1 0 1 977 703 1 274 0 1 -1 703 274 2 155 1 -1 3 274 155 1 119 -1 3 -4 155 119 1 36 3 -4 7 119 36 3 11 -4 7 -25 36 11 3 3 7 -25 82 11 3 3 2 -25 82 -271 3 2 1 1 82 -271 353 2 1 2 0 -271 353 -977
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
= (33 × 188561 × 164 +
186 × 270629 × 9 +
766 × 53461 × 353) mod 52231397
= 50927868 (mod 52231397)
So our answer is 50927868 (mod 52231397).
Verification
So we found that x ≡ 50927868
If this is correct, then the following statements (i.e. the original equations) are true:
755x (mod 277) ≡ 816 (mod 277)
246x (mod 193) ≡ 401 (mod 193)
86x (mod 977) ≡ 417 (mod 977)
Let's see whether that's indeed the case if we use x ≡ 50927868.