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 |
|---|---|---|---|---|---|---|
| 247 | 150 | 1 | 97 | 0 | 1 | -1 |
| 150 | 97 | 1 | 53 | 1 | -1 | 2 |
| 97 | 53 | 1 | 44 | -1 | 2 | -3 |
| 53 | 44 | 1 | 9 | 2 | -3 | 5 |
| 44 | 9 | 4 | 8 | -3 | 5 | -23 |
| 9 | 8 | 1 | 1 | 5 | -23 | 28 |
| 8 | 1 | 8 | 0 | -23 | 28 | -247 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 443 | 88 | 5 | 3 | 0 | 1 | -5 |
| 88 | 3 | 29 | 1 | 1 | -5 | 146 |
| 3 | 1 | 3 | 0 | -5 | 146 | -443 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 467 | 262 | 1 | 205 | 0 | 1 | -1 |
| 262 | 205 | 1 | 57 | 1 | -1 | 2 |
| 205 | 57 | 3 | 34 | -1 | 2 | -7 |
| 57 | 34 | 1 | 23 | 2 | -7 | 9 |
| 34 | 23 | 1 | 11 | -7 | 9 | -16 |
| 23 | 11 | 2 | 1 | 9 | -16 | 41 |
| 11 | 1 | 11 | 0 | -16 | 41 | -467 |
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 ≡ 44 × 150-1 (mod 247) ≡ 44 × 28 (mod 247) ≡ 244 (mod 247)x ≡ 832 × 88-1 (mod 443) ≡ 832 × 146 (mod 443) ≡ 90 (mod 443)x ≡ 307 × 262-1 (mod 467) ≡ 307 × 41 (mod 467) ≡ 445 (mod 467)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 244 (mod 247)
x ≡ 90 (mod 443)
x ≡ 445 (mod 467)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 247 × 443 × 467 = 51099607 - We calculate the numbers M1 to M3
M1=M/m1=51099607/247=206881,M2=M/m2=51099607/443=115349,M3=M/m3=51099607/467=109421 - 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 -40 mod 247 ≡ 207n b q r t1 t2 t3 247 206881 0 247 0 1 0 206881 247 837 142 1 0 1 247 142 1 105 0 1 -1 142 105 1 37 1 -1 2 105 37 2 31 -1 2 -5 37 31 1 6 2 -5 7 31 6 5 1 -5 7 -40 6 1 6 0 7 -40 247
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -173 mod 443 ≡ 270n b q r t1 t2 t3 443 115349 0 443 0 1 0 115349 443 260 169 1 0 1 443 169 2 105 0 1 -2 169 105 1 64 1 -2 3 105 64 1 41 -2 3 -5 64 41 1 23 3 -5 8 41 23 1 18 -5 8 -13 23 18 1 5 8 -13 21 18 5 3 3 -13 21 -76 5 3 1 2 21 -76 97 3 2 1 1 -76 97 -173 2 1 2 0 97 -173 443
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -209 mod 467 ≡ 258n b q r t1 t2 t3 467 109421 0 467 0 1 0 109421 467 234 143 1 0 1 467 143 3 38 0 1 -3 143 38 3 29 1 -3 10 38 29 1 9 -3 10 -13 29 9 3 2 10 -13 49 9 2 4 1 -13 49 -209 2 1 2 0 49 -209 467
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
= (244 × 206881 × 207 +
90 × 115349 × 270 +
445 × 109421 × 258) mod 51099607
= 9449723 (mod 51099607)
So our answer is 9449723 (mod 51099607).
Verification
So we found that x ≡ 9449723
If this is correct, then the following statements (i.e. the original equations) are true:
150x (mod 247) ≡ 44 (mod 247)
88x (mod 443) ≡ 832 (mod 443)
262x (mod 467) ≡ 307 (mod 467)
Let's see whether that's indeed the case if we use x ≡ 9449723.