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 |
|---|---|---|---|---|---|---|
| 295 | 89 | 3 | 28 | 0 | 1 | -3 |
| 89 | 28 | 3 | 5 | 1 | -3 | 10 |
| 28 | 5 | 5 | 3 | -3 | 10 | -53 |
| 5 | 3 | 1 | 2 | 10 | -53 | 63 |
| 3 | 2 | 1 | 1 | -53 | 63 | -116 |
| 2 | 1 | 2 | 0 | 63 | -116 | 295 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 811 | 917 | 0 | 811 | 0 | 1 | 0 |
| 917 | 811 | 1 | 106 | 1 | 0 | 1 |
| 811 | 106 | 7 | 69 | 0 | 1 | -7 |
| 106 | 69 | 1 | 37 | 1 | -7 | 8 |
| 69 | 37 | 1 | 32 | -7 | 8 | -15 |
| 37 | 32 | 1 | 5 | 8 | -15 | 23 |
| 32 | 5 | 6 | 2 | -15 | 23 | -153 |
| 5 | 2 | 2 | 1 | 23 | -153 | 329 |
| 2 | 1 | 2 | 0 | -153 | 329 | -811 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 347 | 724 | 0 | 347 | 0 | 1 | 0 |
| 724 | 347 | 2 | 30 | 1 | 0 | 1 |
| 347 | 30 | 11 | 17 | 0 | 1 | -11 |
| 30 | 17 | 1 | 13 | 1 | -11 | 12 |
| 17 | 13 | 1 | 4 | -11 | 12 | -23 |
| 13 | 4 | 3 | 1 | 12 | -23 | 81 |
| 4 | 1 | 4 | 0 | -23 | 81 | -347 |
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 ≡ 659 × 89-1 (mod 295) ≡ 659 × 179 (mod 295) ≡ 256 (mod 295)x ≡ 138 × 917-1 (mod 811) ≡ 138 × 329 (mod 811) ≡ 797 (mod 811)x ≡ 914 × 724-1 (mod 347) ≡ 914 × 81 (mod 347) ≡ 123 (mod 347)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 256 (mod 295)
x ≡ 797 (mod 811)
x ≡ 123 (mod 347)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 295 × 811 × 347 = 83018015 - We calculate the numbers M1 to M3
M1=M/m1=83018015/295=281417,M2=M/m2=83018015/811=102365,M3=M/m3=83018015/347=239245 - 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 68 mod 295 ≡ 68n b q r t1 t2 t3 295 281417 0 295 0 1 0 281417 295 953 282 1 0 1 295 282 1 13 0 1 -1 282 13 21 9 1 -1 22 13 9 1 4 -1 22 -23 9 4 2 1 22 -23 68 4 1 4 0 -23 68 -295
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -222 mod 811 ≡ 589n b q r t1 t2 t3 811 102365 0 811 0 1 0 102365 811 126 179 1 0 1 811 179 4 95 0 1 -4 179 95 1 84 1 -4 5 95 84 1 11 -4 5 -9 84 11 7 7 5 -9 68 11 7 1 4 -9 68 -77 7 4 1 3 68 -77 145 4 3 1 1 -77 145 -222 3 1 3 0 145 -222 811
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 15 mod 347 ≡ 15n b q r t1 t2 t3 347 239245 0 347 0 1 0 239245 347 689 162 1 0 1 347 162 2 23 0 1 -2 162 23 7 1 1 -2 15 23 1 23 0 -2 15 -347
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
= (256 × 281417 × 68 +
797 × 102365 × 589 +
123 × 239245 × 15) mod 83018015
= 13239561 (mod 83018015)
So our answer is 13239561 (mod 83018015).
Verification
So we found that x ≡ 13239561
If this is correct, then the following statements (i.e. the original equations) are true:
89x (mod 295) ≡ 659 (mod 295)
917x (mod 811) ≡ 138 (mod 811)
724x (mod 347) ≡ 914 (mod 347)
Let's see whether that's indeed the case if we use x ≡ 13239561.