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" .
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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 |
|---|---|---|---|---|---|---|
| 179 | 10 | 17 | 9 | 0 | 1 | -17 |
| 10 | 9 | 1 | 1 | 1 | -17 | 18 |
| 9 | 1 | 9 | 0 | -17 | 18 | -179 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 769 | 429 | 1 | 340 | 0 | 1 | -1 |
| 429 | 340 | 1 | 89 | 1 | -1 | 2 |
| 340 | 89 | 3 | 73 | -1 | 2 | -7 |
| 89 | 73 | 1 | 16 | 2 | -7 | 9 |
| 73 | 16 | 4 | 9 | -7 | 9 | -43 |
| 16 | 9 | 1 | 7 | 9 | -43 | 52 |
| 9 | 7 | 1 | 2 | -43 | 52 | -95 |
| 7 | 2 | 3 | 1 | 52 | -95 | 337 |
| 2 | 1 | 2 | 0 | -95 | 337 | -769 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 565 | 143 | 3 | 136 | 0 | 1 | -3 |
| 143 | 136 | 1 | 7 | 1 | -3 | 4 |
| 136 | 7 | 19 | 3 | -3 | 4 | -79 |
| 7 | 3 | 2 | 1 | 4 | -79 | 162 |
| 3 | 1 | 3 | 0 | -79 | 162 | -565 |
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 ≡ 139 × 10-1 (mod 179) ≡ 139 × 18 (mod 179) ≡ 175 (mod 179)x ≡ 206 × 429-1 (mod 769) ≡ 206 × 337 (mod 769) ≡ 212 (mod 769)x ≡ 691 × 143-1 (mod 565) ≡ 691 × 162 (mod 565) ≡ 72 (mod 565)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 175 (mod 179)
x ≡ 212 (mod 769)
x ≡ 72 (mod 565)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 179 × 769 × 565 = 77772815 - We calculate the numbers M1 to M3
M1=M/m1=77772815/179=434485,M2=M/m2=77772815/769=101135,M3=M/m3=77772815/565=137651 - 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 31 mod 179 ≡ 31n b q r t1 t2 t3 179 434485 0 179 0 1 0 434485 179 2427 52 1 0 1 179 52 3 23 0 1 -3 52 23 2 6 1 -3 7 23 6 3 5 -3 7 -24 6 5 1 1 7 -24 31 5 1 5 0 -24 31 -179
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 301 mod 769 ≡ 301n b q r t1 t2 t3 769 101135 0 769 0 1 0 101135 769 131 396 1 0 1 769 396 1 373 0 1 -1 396 373 1 23 1 -1 2 373 23 16 5 -1 2 -33 23 5 4 3 2 -33 134 5 3 1 2 -33 134 -167 3 2 1 1 134 -167 301 2 1 2 0 -167 301 -769
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 246 mod 565 ≡ 246n b q r t1 t2 t3 565 137651 0 565 0 1 0 137651 565 243 356 1 0 1 565 356 1 209 0 1 -1 356 209 1 147 1 -1 2 209 147 1 62 -1 2 -3 147 62 2 23 2 -3 8 62 23 2 16 -3 8 -19 23 16 1 7 8 -19 27 16 7 2 2 -19 27 -73 7 2 3 1 27 -73 246 2 1 2 0 -73 246 -565
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
= (175 × 434485 × 31 +
212 × 101135 × 301 +
72 × 137651 × 246) mod 77772815
= 49496897 (mod 77772815)
So our answer is 49496897 (mod 77772815).
Verification
So we found that x ≡ 49496897
If this is correct, then the following statements (i.e. the original equations) are true:
10x (mod 179) ≡ 139 (mod 179)
429x (mod 769) ≡ 206 (mod 769)
143x (mod 565) ≡ 691 (mod 565)
Let's see whether that's indeed the case if we use x ≡ 49496897.