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 |
|---|---|---|---|---|---|---|
| 503 | 401 | 1 | 102 | 0 | 1 | -1 |
| 401 | 102 | 3 | 95 | 1 | -1 | 4 |
| 102 | 95 | 1 | 7 | -1 | 4 | -5 |
| 95 | 7 | 13 | 4 | 4 | -5 | 69 |
| 7 | 4 | 1 | 3 | -5 | 69 | -74 |
| 4 | 3 | 1 | 1 | 69 | -74 | 143 |
| 3 | 1 | 3 | 0 | -74 | 143 | -503 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 517 | 171 | 3 | 4 | 0 | 1 | -3 |
| 171 | 4 | 42 | 3 | 1 | -3 | 127 |
| 4 | 3 | 1 | 1 | -3 | 127 | -130 |
| 3 | 1 | 3 | 0 | 127 | -130 | 517 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 379 | 603 | 0 | 379 | 0 | 1 | 0 |
| 603 | 379 | 1 | 224 | 1 | 0 | 1 |
| 379 | 224 | 1 | 155 | 0 | 1 | -1 |
| 224 | 155 | 1 | 69 | 1 | -1 | 2 |
| 155 | 69 | 2 | 17 | -1 | 2 | -5 |
| 69 | 17 | 4 | 1 | 2 | -5 | 22 |
| 17 | 1 | 17 | 0 | -5 | 22 | -379 |
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 ≡ 528 × 401-1 (mod 503) ≡ 528 × 143 (mod 503) ≡ 54 (mod 503)x ≡ 143 × 171-1 (mod 517) ≡ 143 × 387 (mod 517) ≡ 22 (mod 517)x ≡ 471 × 603-1 (mod 379) ≡ 471 × 22 (mod 379) ≡ 129 (mod 379)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 54 (mod 503)
x ≡ 22 (mod 517)
x ≡ 129 (mod 379)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 503 × 517 × 379 = 98559329 - We calculate the numbers M1 to M3
M1=M/m1=98559329/503=195943,M2=M/m2=98559329/517=190637,M3=M/m3=98559329/379=260051 - 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 -195 mod 503 ≡ 308n b q r t1 t2 t3 503 195943 0 503 0 1 0 195943 503 389 276 1 0 1 503 276 1 227 0 1 -1 276 227 1 49 1 -1 2 227 49 4 31 -1 2 -9 49 31 1 18 2 -9 11 31 18 1 13 -9 11 -20 18 13 1 5 11 -20 31 13 5 2 3 -20 31 -82 5 3 1 2 31 -82 113 3 2 1 1 -82 113 -195 2 1 2 0 113 -195 503
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 19 mod 517 ≡ 19n b q r t1 t2 t3 517 190637 0 517 0 1 0 190637 517 368 381 1 0 1 517 381 1 136 0 1 -1 381 136 2 109 1 -1 3 136 109 1 27 -1 3 -4 109 27 4 1 3 -4 19 27 1 27 0 -4 19 -517
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 133 mod 379 ≡ 133n b q r t1 t2 t3 379 260051 0 379 0 1 0 260051 379 686 57 1 0 1 379 57 6 37 0 1 -6 57 37 1 20 1 -6 7 37 20 1 17 -6 7 -13 20 17 1 3 7 -13 20 17 3 5 2 -13 20 -113 3 2 1 1 20 -113 133 2 1 2 0 -113 133 -379
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
= (54 × 195943 × 308 +
22 × 190637 × 19 +
129 × 260051 × 133) mod 98559329
= 14118258 (mod 98559329)
So our answer is 14118258 (mod 98559329).
Verification
So we found that x ≡ 14118258
If this is correct, then the following statements (i.e. the original equations) are true:
401x (mod 503) ≡ 528 (mod 503)
171x (mod 517) ≡ 143 (mod 517)
603x (mod 379) ≡ 471 (mod 379)
Let's see whether that's indeed the case if we use x ≡ 14118258.