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 |
|---|---|---|---|---|---|---|
| 731 | 492 | 1 | 239 | 0 | 1 | -1 |
| 492 | 239 | 2 | 14 | 1 | -1 | 3 |
| 239 | 14 | 17 | 1 | -1 | 3 | -52 |
| 14 | 1 | 14 | 0 | 3 | -52 | 731 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 589 | 293 | 2 | 3 | 0 | 1 | -2 |
| 293 | 3 | 97 | 2 | 1 | -2 | 195 |
| 3 | 2 | 1 | 1 | -2 | 195 | -197 |
| 2 | 1 | 2 | 0 | 195 | -197 | 589 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 173 | 310 | 0 | 173 | 0 | 1 | 0 |
| 310 | 173 | 1 | 137 | 1 | 0 | 1 |
| 173 | 137 | 1 | 36 | 0 | 1 | -1 |
| 137 | 36 | 3 | 29 | 1 | -1 | 4 |
| 36 | 29 | 1 | 7 | -1 | 4 | -5 |
| 29 | 7 | 4 | 1 | 4 | -5 | 24 |
| 7 | 1 | 7 | 0 | -5 | 24 | -173 |
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 ≡ 270 × 492-1 (mod 731) ≡ 270 × 679 (mod 731) ≡ 580 (mod 731)x ≡ 702 × 293-1 (mod 589) ≡ 702 × 392 (mod 589) ≡ 121 (mod 589)x ≡ 751 × 310-1 (mod 173) ≡ 751 × 24 (mod 173) ≡ 32 (mod 173)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 580 (mod 731)
x ≡ 121 (mod 589)
x ≡ 32 (mod 173)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 731 × 589 × 173 = 74486707 - We calculate the numbers M1 to M3
M1=M/m1=74486707/731=101897,M2=M/m2=74486707/589=126463,M3=M/m3=74486707/173=430559 - 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 33 mod 731 ≡ 33n b q r t1 t2 t3 731 101897 0 731 0 1 0 101897 731 139 288 1 0 1 731 288 2 155 0 1 -2 288 155 1 133 1 -2 3 155 133 1 22 -2 3 -5 133 22 6 1 3 -5 33 22 1 22 0 -5 33 -731
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 113 mod 589 ≡ 113n b q r t1 t2 t3 589 126463 0 589 0 1 0 126463 589 214 417 1 0 1 589 417 1 172 0 1 -1 417 172 2 73 1 -1 3 172 73 2 26 -1 3 -7 73 26 2 21 3 -7 17 26 21 1 5 -7 17 -24 21 5 4 1 17 -24 113 5 1 5 0 -24 113 -589
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -41 mod 173 ≡ 132n b q r t1 t2 t3 173 430559 0 173 0 1 0 430559 173 2488 135 1 0 1 173 135 1 38 0 1 -1 135 38 3 21 1 -1 4 38 21 1 17 -1 4 -5 21 17 1 4 4 -5 9 17 4 4 1 -5 9 -41 4 1 4 0 9 -41 173
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
= (580 × 101897 × 33 +
121 × 126463 × 113 +
32 × 430559 × 132) mod 74486707
= 60588784 (mod 74486707)
So our answer is 60588784 (mod 74486707).
Verification
So we found that x ≡ 60588784
If this is correct, then the following statements (i.e. the original equations) are true:
492x (mod 731) ≡ 270 (mod 731)
293x (mod 589) ≡ 702 (mod 589)
310x (mod 173) ≡ 751 (mod 173)
Let's see whether that's indeed the case if we use x ≡ 60588784.