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" .
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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 |
|---|---|---|---|---|---|---|
| 137 | 210 | 0 | 137 | 0 | 1 | 0 |
| 210 | 137 | 1 | 73 | 1 | 0 | 1 |
| 137 | 73 | 1 | 64 | 0 | 1 | -1 |
| 73 | 64 | 1 | 9 | 1 | -1 | 2 |
| 64 | 9 | 7 | 1 | -1 | 2 | -15 |
| 9 | 1 | 9 | 0 | 2 | -15 | 137 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 515 | 23 | 22 | 9 | 0 | 1 | -22 |
| 23 | 9 | 2 | 5 | 1 | -22 | 45 |
| 9 | 5 | 1 | 4 | -22 | 45 | -67 |
| 5 | 4 | 1 | 1 | 45 | -67 | 112 |
| 4 | 1 | 4 | 0 | -67 | 112 | -515 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 544 | 7 | 77 | 5 | 0 | 1 | -77 |
| 7 | 5 | 1 | 2 | 1 | -77 | 78 |
| 5 | 2 | 2 | 1 | -77 | 78 | -233 |
| 2 | 1 | 2 | 0 | 78 | -233 | 544 |
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 ≡ 781 × 210-1 (mod 137) ≡ 781 × 122 (mod 137) ≡ 67 (mod 137)x ≡ 623 × 23-1 (mod 515) ≡ 623 × 112 (mod 515) ≡ 251 (mod 515)x ≡ 961 × 7-1 (mod 544) ≡ 961 × 311 (mod 544) ≡ 215 (mod 544)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 67 (mod 137)
x ≡ 251 (mod 515)
x ≡ 215 (mod 544)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 137 × 515 × 544 = 38381920 - We calculate the numbers M1 to M3
M1=M/m1=38381920/137=280160,M2=M/m2=38381920/515=74528,M3=M/m3=38381920/544=70555 - 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 -55 mod 137 ≡ 82n b q r t1 t2 t3 137 280160 0 137 0 1 0 280160 137 2044 132 1 0 1 137 132 1 5 0 1 -1 132 5 26 2 1 -1 27 5 2 2 1 -1 27 -55 2 1 2 0 27 -55 137
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 7 mod 515 ≡ 7n b q r t1 t2 t3 515 74528 0 515 0 1 0 74528 515 144 368 1 0 1 515 368 1 147 0 1 -1 368 147 2 74 1 -1 3 147 74 1 73 -1 3 -4 74 73 1 1 3 -4 7 73 1 73 0 -4 7 -515
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 211 mod 544 ≡ 211n b q r t1 t2 t3 544 70555 0 544 0 1 0 70555 544 129 379 1 0 1 544 379 1 165 0 1 -1 379 165 2 49 1 -1 3 165 49 3 18 -1 3 -10 49 18 2 13 3 -10 23 18 13 1 5 -10 23 -33 13 5 2 3 23 -33 89 5 3 1 2 -33 89 -122 3 2 1 1 89 -122 211 2 1 2 0 -122 211 -544
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
= (67 × 280160 × 82 +
251 × 74528 × 7 +
215 × 70555 × 211) mod 38381920
= 34750391 (mod 38381920)
So our answer is 34750391 (mod 38381920).
Verification
So we found that x ≡ 34750391
If this is correct, then the following statements (i.e. the original equations) are true:
210x (mod 137) ≡ 781 (mod 137)
23x (mod 515) ≡ 623 (mod 515)
7x (mod 544) ≡ 961 (mod 544)
Let's see whether that's indeed the case if we use x ≡ 34750391.