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 |
|---|---|---|---|---|---|---|
| 491 | 933 | 0 | 491 | 0 | 1 | 0 |
| 933 | 491 | 1 | 442 | 1 | 0 | 1 |
| 491 | 442 | 1 | 49 | 0 | 1 | -1 |
| 442 | 49 | 9 | 1 | 1 | -1 | 10 |
| 49 | 1 | 49 | 0 | -1 | 10 | -491 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 383 | 43 | 8 | 39 | 0 | 1 | -8 |
| 43 | 39 | 1 | 4 | 1 | -8 | 9 |
| 39 | 4 | 9 | 3 | -8 | 9 | -89 |
| 4 | 3 | 1 | 1 | 9 | -89 | 98 |
| 3 | 1 | 3 | 0 | -89 | 98 | -383 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 179 | 167 | 1 | 12 | 0 | 1 | -1 |
| 167 | 12 | 13 | 11 | 1 | -1 | 14 |
| 12 | 11 | 1 | 1 | -1 | 14 | -15 |
| 11 | 1 | 11 | 0 | 14 | -15 | 179 |
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 ≡ 640 × 933-1 (mod 491) ≡ 640 × 10 (mod 491) ≡ 17 (mod 491)x ≡ 293 × 43-1 (mod 383) ≡ 293 × 98 (mod 383) ≡ 372 (mod 383)x ≡ 673 × 167-1 (mod 179) ≡ 673 × 164 (mod 179) ≡ 108 (mod 179)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 17 (mod 491)
x ≡ 372 (mod 383)
x ≡ 108 (mod 179)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 491 × 383 × 179 = 33661487 - We calculate the numbers M1 to M3
M1=M/m1=33661487/491=68557,M2=M/m2=33661487/383=87889,M3=M/m3=33661487/179=188053 - 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 110 mod 491 ≡ 110n b q r t1 t2 t3 491 68557 0 491 0 1 0 68557 491 139 308 1 0 1 491 308 1 183 0 1 -1 308 183 1 125 1 -1 2 183 125 1 58 -1 2 -3 125 58 2 9 2 -3 8 58 9 6 4 -3 8 -51 9 4 2 1 8 -51 110 4 1 4 0 -51 110 -491
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 141 mod 383 ≡ 141n b q r t1 t2 t3 383 87889 0 383 0 1 0 87889 383 229 182 1 0 1 383 182 2 19 0 1 -2 182 19 9 11 1 -2 19 19 11 1 8 -2 19 -21 11 8 1 3 19 -21 40 8 3 2 2 -21 40 -101 3 2 1 1 40 -101 141 2 1 2 0 -101 141 -383
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 73 mod 179 ≡ 73n b q r t1 t2 t3 179 188053 0 179 0 1 0 188053 179 1050 103 1 0 1 179 103 1 76 0 1 -1 103 76 1 27 1 -1 2 76 27 2 22 -1 2 -5 27 22 1 5 2 -5 7 22 5 4 2 -5 7 -33 5 2 2 1 7 -33 73 2 1 2 0 -33 73 -179
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
= (17 × 68557 × 110 +
372 × 87889 × 141 +
108 × 188053 × 73) mod 33661487
= 27051662 (mod 33661487)
So our answer is 27051662 (mod 33661487).
Verification
So we found that x ≡ 27051662
If this is correct, then the following statements (i.e. the original equations) are true:
933x (mod 491) ≡ 640 (mod 491)
43x (mod 383) ≡ 293 (mod 383)
167x (mod 179) ≡ 673 (mod 179)
Let's see whether that's indeed the case if we use x ≡ 27051662.