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 |
|---|---|---|---|---|---|---|
| 187 | 49 | 3 | 40 | 0 | 1 | -3 |
| 49 | 40 | 1 | 9 | 1 | -3 | 4 |
| 40 | 9 | 4 | 4 | -3 | 4 | -19 |
| 9 | 4 | 2 | 1 | 4 | -19 | 42 |
| 4 | 1 | 4 | 0 | -19 | 42 | -187 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 974 | 719 | 1 | 255 | 0 | 1 | -1 |
| 719 | 255 | 2 | 209 | 1 | -1 | 3 |
| 255 | 209 | 1 | 46 | -1 | 3 | -4 |
| 209 | 46 | 4 | 25 | 3 | -4 | 19 |
| 46 | 25 | 1 | 21 | -4 | 19 | -23 |
| 25 | 21 | 1 | 4 | 19 | -23 | 42 |
| 21 | 4 | 5 | 1 | -23 | 42 | -233 |
| 4 | 1 | 4 | 0 | 42 | -233 | 974 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 757 | 454 | 1 | 303 | 0 | 1 | -1 |
| 454 | 303 | 1 | 151 | 1 | -1 | 2 |
| 303 | 151 | 2 | 1 | -1 | 2 | -5 |
| 151 | 1 | 151 | 0 | 2 | -5 | 757 |
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 ≡ 699 × 49-1 (mod 187) ≡ 699 × 42 (mod 187) ≡ 186 (mod 187)x ≡ 711 × 719-1 (mod 974) ≡ 711 × 741 (mod 974) ≡ 891 (mod 974)x ≡ 55 × 454-1 (mod 757) ≡ 55 × 752 (mod 757) ≡ 482 (mod 757)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 186 (mod 187)
x ≡ 891 (mod 974)
x ≡ 482 (mod 757)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 187 × 974 × 757 = 137878466 - We calculate the numbers M1 to M3
M1=M/m1=137878466/187=737318,M2=M/m2=137878466/974=141559,M3=M/m3=137878466/757=182138 - 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 65 mod 187 ≡ 65n b q r t1 t2 t3 187 737318 0 187 0 1 0 737318 187 3942 164 1 0 1 187 164 1 23 0 1 -1 164 23 7 3 1 -1 8 23 3 7 2 -1 8 -57 3 2 1 1 8 -57 65 2 1 2 0 -57 65 -187
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 225 mod 974 ≡ 225n b q r t1 t2 t3 974 141559 0 974 0 1 0 141559 974 145 329 1 0 1 974 329 2 316 0 1 -2 329 316 1 13 1 -2 3 316 13 24 4 -2 3 -74 13 4 3 1 3 -74 225 4 1 4 0 -74 225 -974
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 319 mod 757 ≡ 319n b q r t1 t2 t3 757 182138 0 757 0 1 0 182138 757 240 458 1 0 1 757 458 1 299 0 1 -1 458 299 1 159 1 -1 2 299 159 1 140 -1 2 -3 159 140 1 19 2 -3 5 140 19 7 7 -3 5 -38 19 7 2 5 5 -38 81 7 5 1 2 -38 81 -119 5 2 2 1 81 -119 319 2 1 2 0 -119 319 -757
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
= (186 × 737318 × 65 +
891 × 141559 × 225 +
482 × 182138 × 319) mod 137878466
= 81875331 (mod 137878466)
So our answer is 81875331 (mod 137878466).
Verification
So we found that x ≡ 81875331
If this is correct, then the following statements (i.e. the original equations) are true:
49x (mod 187) ≡ 699 (mod 187)
719x (mod 974) ≡ 711 (mod 974)
454x (mod 757) ≡ 55 (mod 757)
Let's see whether that's indeed the case if we use x ≡ 81875331.