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 |
|---|---|---|---|---|---|---|
| 967 | 298 | 3 | 73 | 0 | 1 | -3 |
| 298 | 73 | 4 | 6 | 1 | -3 | 13 |
| 73 | 6 | 12 | 1 | -3 | 13 | -159 |
| 6 | 1 | 6 | 0 | 13 | -159 | 967 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 158 | 23 | 6 | 20 | 0 | 1 | -6 |
| 23 | 20 | 1 | 3 | 1 | -6 | 7 |
| 20 | 3 | 6 | 2 | -6 | 7 | -48 |
| 3 | 2 | 1 | 1 | 7 | -48 | 55 |
| 2 | 1 | 2 | 0 | -48 | 55 | -158 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 971 | 59 | 16 | 27 | 0 | 1 | -16 |
| 59 | 27 | 2 | 5 | 1 | -16 | 33 |
| 27 | 5 | 5 | 2 | -16 | 33 | -181 |
| 5 | 2 | 2 | 1 | 33 | -181 | 395 |
| 2 | 1 | 2 | 0 | -181 | 395 | -971 |
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 ≡ 978 × 298-1 (mod 967) ≡ 978 × 808 (mod 967) ≡ 185 (mod 967)x ≡ 41 × 23-1 (mod 158) ≡ 41 × 55 (mod 158) ≡ 43 (mod 158)x ≡ 100 × 59-1 (mod 971) ≡ 100 × 395 (mod 971) ≡ 660 (mod 971)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 185 (mod 967)
x ≡ 43 (mod 158)
x ≡ 660 (mod 971)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 967 × 158 × 971 = 148355206 - We calculate the numbers M1 to M3
M1=M/m1=148355206/967=153418,M2=M/m2=148355206/158=938957,M3=M/m3=148355206/971=152786 - 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 -280 mod 967 ≡ 687n b q r t1 t2 t3 967 153418 0 967 0 1 0 153418 967 158 632 1 0 1 967 632 1 335 0 1 -1 632 335 1 297 1 -1 2 335 297 1 38 -1 2 -3 297 38 7 31 2 -3 23 38 31 1 7 -3 23 -26 31 7 4 3 23 -26 127 7 3 2 1 -26 127 -280 3 1 3 0 127 -280 967
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -47 mod 158 ≡ 111n b q r t1 t2 t3 158 938957 0 158 0 1 0 938957 158 5942 121 1 0 1 158 121 1 37 0 1 -1 121 37 3 10 1 -1 4 37 10 3 7 -1 4 -13 10 7 1 3 4 -13 17 7 3 2 1 -13 17 -47 3 1 3 0 17 -47 158
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -401 mod 971 ≡ 570n b q r t1 t2 t3 971 152786 0 971 0 1 0 152786 971 157 339 1 0 1 971 339 2 293 0 1 -2 339 293 1 46 1 -2 3 293 46 6 17 -2 3 -20 46 17 2 12 3 -20 43 17 12 1 5 -20 43 -63 12 5 2 2 43 -63 169 5 2 2 1 -63 169 -401 2 1 2 0 169 -401 971
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
= (185 × 153418 × 687 +
43 × 938957 × 111 +
660 × 152786 × 570) mod 148355206
= 11387577 (mod 148355206)
So our answer is 11387577 (mod 148355206).
Verification
So we found that x ≡ 11387577
If this is correct, then the following statements (i.e. the original equations) are true:
298x (mod 967) ≡ 978 (mod 967)
23x (mod 158) ≡ 41 (mod 158)
59x (mod 971) ≡ 100 (mod 971)
Let's see whether that's indeed the case if we use x ≡ 11387577.