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 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 |
|---|---|---|---|---|---|---|
| 179 | 141 | 1 | 38 | 0 | 1 | -1 |
| 141 | 38 | 3 | 27 | 1 | -1 | 4 |
| 38 | 27 | 1 | 11 | -1 | 4 | -5 |
| 27 | 11 | 2 | 5 | 4 | -5 | 14 |
| 11 | 5 | 2 | 1 | -5 | 14 | -33 |
| 5 | 1 | 5 | 0 | 14 | -33 | 179 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 643 | 946 | 0 | 643 | 0 | 1 | 0 |
| 946 | 643 | 1 | 303 | 1 | 0 | 1 |
| 643 | 303 | 2 | 37 | 0 | 1 | -2 |
| 303 | 37 | 8 | 7 | 1 | -2 | 17 |
| 37 | 7 | 5 | 2 | -2 | 17 | -87 |
| 7 | 2 | 3 | 1 | 17 | -87 | 278 |
| 2 | 1 | 2 | 0 | -87 | 278 | -643 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 773 | 55 | 14 | 3 | 0 | 1 | -14 |
| 55 | 3 | 18 | 1 | 1 | -14 | 253 |
| 3 | 1 | 3 | 0 | -14 | 253 | -773 |
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 ≡ 759 × 141-1 (mod 179) ≡ 759 × 146 (mod 179) ≡ 13 (mod 179)x ≡ 602 × 946-1 (mod 643) ≡ 602 × 278 (mod 643) ≡ 176 (mod 643)x ≡ 177 × 55-1 (mod 773) ≡ 177 × 253 (mod 773) ≡ 720 (mod 773)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 13 (mod 179)
x ≡ 176 (mod 643)
x ≡ 720 (mod 773)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 179 × 643 × 773 = 88969981 - We calculate the numbers M1 to M3
M1=M/m1=88969981/179=497039,M2=M/m2=88969981/643=138367,M3=M/m3=88969981/773=115097 - 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 61 mod 179 ≡ 61n b q r t1 t2 t3 179 497039 0 179 0 1 0 497039 179 2776 135 1 0 1 179 135 1 44 0 1 -1 135 44 3 3 1 -1 4 44 3 14 2 -1 4 -57 3 2 1 1 4 -57 61 2 1 2 0 -57 61 -179
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -195 mod 643 ≡ 448n b q r t1 t2 t3 643 138367 0 643 0 1 0 138367 643 215 122 1 0 1 643 122 5 33 0 1 -5 122 33 3 23 1 -5 16 33 23 1 10 -5 16 -21 23 10 2 3 16 -21 58 10 3 3 1 -21 58 -195 3 1 3 0 58 -195 643
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -29 mod 773 ≡ 744n b q r t1 t2 t3 773 115097 0 773 0 1 0 115097 773 148 693 1 0 1 773 693 1 80 0 1 -1 693 80 8 53 1 -1 9 80 53 1 27 -1 9 -10 53 27 1 26 9 -10 19 27 26 1 1 -10 19 -29 26 1 26 0 19 -29 773
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
= (13 × 497039 × 61 +
176 × 138367 × 448 +
720 × 115097 × 744) mod 88969981
= 3889683 (mod 88969981)
So our answer is 3889683 (mod 88969981).
Verification
So we found that x ≡ 3889683
If this is correct, then the following statements (i.e. the original equations) are true:
141x (mod 179) ≡ 759 (mod 179)
946x (mod 643) ≡ 602 (mod 643)
55x (mod 773) ≡ 177 (mod 773)
Let's see whether that's indeed the case if we use x ≡ 3889683.