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 |
|---|---|---|---|---|---|---|
| 223 | 918 | 0 | 223 | 0 | 1 | 0 |
| 918 | 223 | 4 | 26 | 1 | 0 | 1 |
| 223 | 26 | 8 | 15 | 0 | 1 | -8 |
| 26 | 15 | 1 | 11 | 1 | -8 | 9 |
| 15 | 11 | 1 | 4 | -8 | 9 | -17 |
| 11 | 4 | 2 | 3 | 9 | -17 | 43 |
| 4 | 3 | 1 | 1 | -17 | 43 | -60 |
| 3 | 1 | 3 | 0 | 43 | -60 | 223 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 535 | 476 | 1 | 59 | 0 | 1 | -1 |
| 476 | 59 | 8 | 4 | 1 | -1 | 9 |
| 59 | 4 | 14 | 3 | -1 | 9 | -127 |
| 4 | 3 | 1 | 1 | 9 | -127 | 136 |
| 3 | 1 | 3 | 0 | -127 | 136 | -535 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 989 | 271 | 3 | 176 | 0 | 1 | -3 |
| 271 | 176 | 1 | 95 | 1 | -3 | 4 |
| 176 | 95 | 1 | 81 | -3 | 4 | -7 |
| 95 | 81 | 1 | 14 | 4 | -7 | 11 |
| 81 | 14 | 5 | 11 | -7 | 11 | -62 |
| 14 | 11 | 1 | 3 | 11 | -62 | 73 |
| 11 | 3 | 3 | 2 | -62 | 73 | -281 |
| 3 | 2 | 1 | 1 | 73 | -281 | 354 |
| 2 | 1 | 2 | 0 | -281 | 354 | -989 |
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 ≡ 190 × 918-1 (mod 223) ≡ 190 × 163 (mod 223) ≡ 196 (mod 223)x ≡ 8 × 476-1 (mod 535) ≡ 8 × 136 (mod 535) ≡ 18 (mod 535)x ≡ 733 × 271-1 (mod 989) ≡ 733 × 354 (mod 989) ≡ 364 (mod 989)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 196 (mod 223)
x ≡ 18 (mod 535)
x ≡ 364 (mod 989)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 223 × 535 × 989 = 117992645 - We calculate the numbers M1 to M3
M1=M/m1=117992645/223=529115,M2=M/m2=117992645/535=220547,M3=M/m3=117992645/989=119305 - 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 108 mod 223 ≡ 108n b q r t1 t2 t3 223 529115 0 223 0 1 0 529115 223 2372 159 1 0 1 223 159 1 64 0 1 -1 159 64 2 31 1 -1 3 64 31 2 2 -1 3 -7 31 2 15 1 3 -7 108 2 1 2 0 -7 108 -223
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 198 mod 535 ≡ 198n b q r t1 t2 t3 535 220547 0 535 0 1 0 220547 535 412 127 1 0 1 535 127 4 27 0 1 -4 127 27 4 19 1 -4 17 27 19 1 8 -4 17 -21 19 8 2 3 17 -21 59 8 3 2 2 -21 59 -139 3 2 1 1 59 -139 198 2 1 2 0 -139 198 -535
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 144 mod 989 ≡ 144n b q r t1 t2 t3 989 119305 0 989 0 1 0 119305 989 120 625 1 0 1 989 625 1 364 0 1 -1 625 364 1 261 1 -1 2 364 261 1 103 -1 2 -3 261 103 2 55 2 -3 8 103 55 1 48 -3 8 -11 55 48 1 7 8 -11 19 48 7 6 6 -11 19 -125 7 6 1 1 19 -125 144 6 1 6 0 -125 144 -989
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
= (196 × 529115 × 108 +
18 × 220547 × 198 +
364 × 119305 × 144) mod 117992645
= 68959378 (mod 117992645)
So our answer is 68959378 (mod 117992645).
Verification
So we found that x ≡ 68959378
If this is correct, then the following statements (i.e. the original equations) are true:
918x (mod 223) ≡ 190 (mod 223)
476x (mod 535) ≡ 8 (mod 535)
271x (mod 989) ≡ 733 (mod 989)
Let's see whether that's indeed the case if we use x ≡ 68959378.