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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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 |
|---|---|---|---|---|---|---|
| 991 | 856 | 1 | 135 | 0 | 1 | -1 |
| 856 | 135 | 6 | 46 | 1 | -1 | 7 |
| 135 | 46 | 2 | 43 | -1 | 7 | -15 |
| 46 | 43 | 1 | 3 | 7 | -15 | 22 |
| 43 | 3 | 14 | 1 | -15 | 22 | -323 |
| 3 | 1 | 3 | 0 | 22 | -323 | 991 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 723 | 341 | 2 | 41 | 0 | 1 | -2 |
| 341 | 41 | 8 | 13 | 1 | -2 | 17 |
| 41 | 13 | 3 | 2 | -2 | 17 | -53 |
| 13 | 2 | 6 | 1 | 17 | -53 | 335 |
| 2 | 1 | 2 | 0 | -53 | 335 | -723 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 923 | 154 | 5 | 153 | 0 | 1 | -5 |
| 154 | 153 | 1 | 1 | 1 | -5 | 6 |
| 153 | 1 | 153 | 0 | -5 | 6 | -923 |
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 ≡ 665 × 856-1 (mod 991) ≡ 665 × 668 (mod 991) ≡ 252 (mod 991)x ≡ 980 × 341-1 (mod 723) ≡ 980 × 335 (mod 723) ≡ 58 (mod 723)x ≡ 822 × 154-1 (mod 923) ≡ 822 × 6 (mod 923) ≡ 317 (mod 923)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 252 (mod 991)
x ≡ 58 (mod 723)
x ≡ 317 (mod 923)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 991 × 723 × 923 = 661323039 - We calculate the numbers M1 to M3
M1=M/m1=661323039/991=667329,M2=M/m2=661323039/723=914693,M3=M/m3=661323039/923=716493 - 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 362 mod 991 ≡ 362n b q r t1 t2 t3 991 667329 0 991 0 1 0 667329 991 673 386 1 0 1 991 386 2 219 0 1 -2 386 219 1 167 1 -2 3 219 167 1 52 -2 3 -5 167 52 3 11 3 -5 18 52 11 4 8 -5 18 -77 11 8 1 3 18 -77 95 8 3 2 2 -77 95 -267 3 2 1 1 95 -267 362 2 1 2 0 -267 362 -991
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 332 mod 723 ≡ 332n b q r t1 t2 t3 723 914693 0 723 0 1 0 914693 723 1265 98 1 0 1 723 98 7 37 0 1 -7 98 37 2 24 1 -7 15 37 24 1 13 -7 15 -22 24 13 1 11 15 -22 37 13 11 1 2 -22 37 -59 11 2 5 1 37 -59 332 2 1 2 0 -59 332 -723
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 162 mod 923 ≡ 162n b q r t1 t2 t3 923 716493 0 923 0 1 0 716493 923 776 245 1 0 1 923 245 3 188 0 1 -3 245 188 1 57 1 -3 4 188 57 3 17 -3 4 -15 57 17 3 6 4 -15 49 17 6 2 5 -15 49 -113 6 5 1 1 49 -113 162 5 1 5 0 -113 162 -923
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
= (252 × 667329 × 362 +
58 × 914693 × 332 +
317 × 716493 × 162) mod 661323039
= 214321840 (mod 661323039)
So our answer is 214321840 (mod 661323039).
Verification
So we found that x ≡ 214321840
If this is correct, then the following statements (i.e. the original equations) are true:
856x (mod 991) ≡ 665 (mod 991)
341x (mod 723) ≡ 980 (mod 723)
154x (mod 923) ≡ 822 (mod 923)
Let's see whether that's indeed the case if we use x ≡ 214321840.