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.
Do you want to use more equations? Go ahead and use this button. It adds another row that you can fill in. Not sure what numbers to put in this newly added row? Use the random numbers button again!
Do you have too many rows? Use one of these buttons to remove a row. You can always add a row again using the yellow "Add a row" button.
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 |
|---|---|---|---|---|---|---|
| 359 | 937 | 0 | 359 | 0 | 1 | 0 |
| 937 | 359 | 2 | 219 | 1 | 0 | 1 |
| 359 | 219 | 1 | 140 | 0 | 1 | -1 |
| 219 | 140 | 1 | 79 | 1 | -1 | 2 |
| 140 | 79 | 1 | 61 | -1 | 2 | -3 |
| 79 | 61 | 1 | 18 | 2 | -3 | 5 |
| 61 | 18 | 3 | 7 | -3 | 5 | -18 |
| 18 | 7 | 2 | 4 | 5 | -18 | 41 |
| 7 | 4 | 1 | 3 | -18 | 41 | -59 |
| 4 | 3 | 1 | 1 | 41 | -59 | 100 |
| 3 | 1 | 3 | 0 | -59 | 100 | -359 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 781 | 41 | 19 | 2 | 0 | 1 | -19 |
| 41 | 2 | 20 | 1 | 1 | -19 | 381 |
| 2 | 1 | 2 | 0 | -19 | 381 | -781 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 977 | 137 | 7 | 18 | 0 | 1 | -7 |
| 137 | 18 | 7 | 11 | 1 | -7 | 50 |
| 18 | 11 | 1 | 7 | -7 | 50 | -57 |
| 11 | 7 | 1 | 4 | 50 | -57 | 107 |
| 7 | 4 | 1 | 3 | -57 | 107 | -164 |
| 4 | 3 | 1 | 1 | 107 | -164 | 271 |
| 3 | 1 | 3 | 0 | -164 | 271 | -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 ≡ 360 × 937-1 (mod 359) ≡ 360 × 100 (mod 359) ≡ 100 (mod 359)x ≡ 293 × 41-1 (mod 781) ≡ 293 × 381 (mod 781) ≡ 731 (mod 781)x ≡ 409 × 137-1 (mod 977) ≡ 409 × 271 (mod 977) ≡ 438 (mod 977)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 100 (mod 359)
x ≡ 731 (mod 781)
x ≡ 438 (mod 977)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 359 × 781 × 977 = 273930283 - We calculate the numbers M1 to M3
M1=M/m1=273930283/359=763037,M2=M/m2=273930283/781=350743,M3=M/m3=273930283/977=280379 - 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 82 mod 359 ≡ 82n b q r t1 t2 t3 359 763037 0 359 0 1 0 763037 359 2125 162 1 0 1 359 162 2 35 0 1 -2 162 35 4 22 1 -2 9 35 22 1 13 -2 9 -11 22 13 1 9 9 -11 20 13 9 1 4 -11 20 -31 9 4 2 1 20 -31 82 4 1 4 0 -31 82 -359
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 95 mod 781 ≡ 95n b q r t1 t2 t3 781 350743 0 781 0 1 0 350743 781 449 74 1 0 1 781 74 10 41 0 1 -10 74 41 1 33 1 -10 11 41 33 1 8 -10 11 -21 33 8 4 1 11 -21 95 8 1 8 0 -21 95 -781
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -342 mod 977 ≡ 635n b q r t1 t2 t3 977 280379 0 977 0 1 0 280379 977 286 957 1 0 1 977 957 1 20 0 1 -1 957 20 47 17 1 -1 48 20 17 1 3 -1 48 -49 17 3 5 2 48 -49 293 3 2 1 1 -49 293 -342 2 1 2 0 293 -342 977
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
= (100 × 763037 × 82 +
731 × 350743 × 95 +
438 × 280379 × 635) mod 273930283
= 119670237 (mod 273930283)
So our answer is 119670237 (mod 273930283).
Verification
So we found that x ≡ 119670237
If this is correct, then the following statements (i.e. the original equations) are true:
937x (mod 359) ≡ 360 (mod 359)
41x (mod 781) ≡ 293 (mod 781)
137x (mod 977) ≡ 409 (mod 977)
Let's see whether that's indeed the case if we use x ≡ 119670237.