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 |
|---|---|---|---|---|---|---|
| 775 | 53 | 14 | 33 | 0 | 1 | -14 |
| 53 | 33 | 1 | 20 | 1 | -14 | 15 |
| 33 | 20 | 1 | 13 | -14 | 15 | -29 |
| 20 | 13 | 1 | 7 | 15 | -29 | 44 |
| 13 | 7 | 1 | 6 | -29 | 44 | -73 |
| 7 | 6 | 1 | 1 | 44 | -73 | 117 |
| 6 | 1 | 6 | 0 | -73 | 117 | -775 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 202 | 447 | 0 | 202 | 0 | 1 | 0 |
| 447 | 202 | 2 | 43 | 1 | 0 | 1 |
| 202 | 43 | 4 | 30 | 0 | 1 | -4 |
| 43 | 30 | 1 | 13 | 1 | -4 | 5 |
| 30 | 13 | 2 | 4 | -4 | 5 | -14 |
| 13 | 4 | 3 | 1 | 5 | -14 | 47 |
| 4 | 1 | 4 | 0 | -14 | 47 | -202 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 743 | 331 | 2 | 81 | 0 | 1 | -2 |
| 331 | 81 | 4 | 7 | 1 | -2 | 9 |
| 81 | 7 | 11 | 4 | -2 | 9 | -101 |
| 7 | 4 | 1 | 3 | 9 | -101 | 110 |
| 4 | 3 | 1 | 1 | -101 | 110 | -211 |
| 3 | 1 | 3 | 0 | 110 | -211 | 743 |
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 ≡ 617 × 53-1 (mod 775) ≡ 617 × 117 (mod 775) ≡ 114 (mod 775)x ≡ 253 × 447-1 (mod 202) ≡ 253 × 47 (mod 202) ≡ 175 (mod 202)x ≡ 256 × 331-1 (mod 743) ≡ 256 × 532 (mod 743) ≡ 223 (mod 743)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 114 (mod 775)
x ≡ 175 (mod 202)
x ≡ 223 (mod 743)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 775 × 202 × 743 = 116316650 - We calculate the numbers M1 to M3
M1=M/m1=116316650/775=150086,M2=M/m2=116316650/202=575825,M3=M/m3=116316650/743=156550 - 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 91 mod 775 ≡ 91n b q r t1 t2 t3 775 150086 0 775 0 1 0 150086 775 193 511 1 0 1 775 511 1 264 0 1 -1 511 264 1 247 1 -1 2 264 247 1 17 -1 2 -3 247 17 14 9 2 -3 44 17 9 1 8 -3 44 -47 9 8 1 1 44 -47 91 8 1 8 0 -47 91 -775
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -21 mod 202 ≡ 181n b q r t1 t2 t3 202 575825 0 202 0 1 0 575825 202 2850 125 1 0 1 202 125 1 77 0 1 -1 125 77 1 48 1 -1 2 77 48 1 29 -1 2 -3 48 29 1 19 2 -3 5 29 19 1 10 -3 5 -8 19 10 1 9 5 -8 13 10 9 1 1 -8 13 -21 9 1 9 0 13 -21 202
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -10 mod 743 ≡ 733n b q r t1 t2 t3 743 156550 0 743 0 1 0 156550 743 210 520 1 0 1 743 520 1 223 0 1 -1 520 223 2 74 1 -1 3 223 74 3 1 -1 3 -10 74 1 74 0 3 -10 743
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
= (114 × 150086 × 91 +
175 × 575825 × 181 +
223 × 156550 × 733) mod 116316650
= 22261989 (mod 116316650)
So our answer is 22261989 (mod 116316650).
Verification
So we found that x ≡ 22261989
If this is correct, then the following statements (i.e. the original equations) are true:
53x (mod 775) ≡ 617 (mod 775)
447x (mod 202) ≡ 253 (mod 202)
331x (mod 743) ≡ 256 (mod 743)
Let's see whether that's indeed the case if we use x ≡ 22261989.