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 |
|---|---|---|---|---|---|---|
| 953 | 463 | 2 | 27 | 0 | 1 | -2 |
| 463 | 27 | 17 | 4 | 1 | -2 | 35 |
| 27 | 4 | 6 | 3 | -2 | 35 | -212 |
| 4 | 3 | 1 | 1 | 35 | -212 | 247 |
| 3 | 1 | 3 | 0 | -212 | 247 | -953 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 199 | 964 | 0 | 199 | 0 | 1 | 0 |
| 964 | 199 | 4 | 168 | 1 | 0 | 1 |
| 199 | 168 | 1 | 31 | 0 | 1 | -1 |
| 168 | 31 | 5 | 13 | 1 | -1 | 6 |
| 31 | 13 | 2 | 5 | -1 | 6 | -13 |
| 13 | 5 | 2 | 3 | 6 | -13 | 32 |
| 5 | 3 | 1 | 2 | -13 | 32 | -45 |
| 3 | 2 | 1 | 1 | 32 | -45 | 77 |
| 2 | 1 | 2 | 0 | -45 | 77 | -199 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 571 | 822 | 0 | 571 | 0 | 1 | 0 |
| 822 | 571 | 1 | 251 | 1 | 0 | 1 |
| 571 | 251 | 2 | 69 | 0 | 1 | -2 |
| 251 | 69 | 3 | 44 | 1 | -2 | 7 |
| 69 | 44 | 1 | 25 | -2 | 7 | -9 |
| 44 | 25 | 1 | 19 | 7 | -9 | 16 |
| 25 | 19 | 1 | 6 | -9 | 16 | -25 |
| 19 | 6 | 3 | 1 | 16 | -25 | 91 |
| 6 | 1 | 6 | 0 | -25 | 91 | -571 |
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 ≡ 377 × 463-1 (mod 953) ≡ 377 × 247 (mod 953) ≡ 678 (mod 953)x ≡ 12 × 964-1 (mod 199) ≡ 12 × 77 (mod 199) ≡ 128 (mod 199)x ≡ 857 × 822-1 (mod 571) ≡ 857 × 91 (mod 571) ≡ 331 (mod 571)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 678 (mod 953)
x ≡ 128 (mod 199)
x ≡ 331 (mod 571)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 953 × 199 × 571 = 108288437 - We calculate the numbers M1 to M3
M1=M/m1=108288437/953=113629,M2=M/m2=108288437/199=544163,M3=M/m3=108288437/571=189647 - 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 -176 mod 953 ≡ 777n b q r t1 t2 t3 953 113629 0 953 0 1 0 113629 953 119 222 1 0 1 953 222 4 65 0 1 -4 222 65 3 27 1 -4 13 65 27 2 11 -4 13 -30 27 11 2 5 13 -30 73 11 5 2 1 -30 73 -176 5 1 5 0 73 -176 953
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -80 mod 199 ≡ 119n b q r t1 t2 t3 199 544163 0 199 0 1 0 544163 199 2734 97 1 0 1 199 97 2 5 0 1 -2 97 5 19 2 1 -2 39 5 2 2 1 -2 39 -80 2 1 2 0 39 -80 199
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -236 mod 571 ≡ 335n b q r t1 t2 t3 571 189647 0 571 0 1 0 189647 571 332 75 1 0 1 571 75 7 46 0 1 -7 75 46 1 29 1 -7 8 46 29 1 17 -7 8 -15 29 17 1 12 8 -15 23 17 12 1 5 -15 23 -38 12 5 2 2 23 -38 99 5 2 2 1 -38 99 -236 2 1 2 0 99 -236 571
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
= (678 × 113629 × 777 +
128 × 544163 × 119 +
331 × 189647 × 335) mod 108288437
= 56753734 (mod 108288437)
So our answer is 56753734 (mod 108288437).
Verification
So we found that x ≡ 56753734
If this is correct, then the following statements (i.e. the original equations) are true:
463x (mod 953) ≡ 377 (mod 953)
964x (mod 199) ≡ 12 (mod 199)
822x (mod 571) ≡ 857 (mod 571)
Let's see whether that's indeed the case if we use x ≡ 56753734.