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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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 |
|---|---|---|---|---|---|---|
| 524 | 525 | 0 | 524 | 0 | 1 | 0 |
| 525 | 524 | 1 | 1 | 1 | 0 | 1 |
| 524 | 1 | 524 | 0 | 0 | 1 | -524 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 625 | 504 | 1 | 121 | 0 | 1 | -1 |
| 504 | 121 | 4 | 20 | 1 | -1 | 5 |
| 121 | 20 | 6 | 1 | -1 | 5 | -31 |
| 20 | 1 | 20 | 0 | 5 | -31 | 625 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 169 | 36 | 4 | 25 | 0 | 1 | -4 |
| 36 | 25 | 1 | 11 | 1 | -4 | 5 |
| 25 | 11 | 2 | 3 | -4 | 5 | -14 |
| 11 | 3 | 3 | 2 | 5 | -14 | 47 |
| 3 | 2 | 1 | 1 | -14 | 47 | -61 |
| 2 | 1 | 2 | 0 | 47 | -61 | 169 |
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 ≡ 172 × 525-1 (mod 524) ≡ 172 × 1 (mod 524) ≡ 172 (mod 524)x ≡ 363 × 504-1 (mod 625) ≡ 363 × 594 (mod 625) ≡ 622 (mod 625)x ≡ 246 × 36-1 (mod 169) ≡ 246 × 108 (mod 169) ≡ 35 (mod 169)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 172 (mod 524)
x ≡ 622 (mod 625)
x ≡ 35 (mod 169)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 524 × 625 × 169 = 55347500 - We calculate the numbers M1 to M3
M1=M/m1=55347500/524=105625,M2=M/m2=55347500/625=88556,M3=M/m3=55347500/169=327500 - 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 -47 mod 524 ≡ 477n b q r t1 t2 t3 524 105625 0 524 0 1 0 105625 524 201 301 1 0 1 524 301 1 223 0 1 -1 301 223 1 78 1 -1 2 223 78 2 67 -1 2 -5 78 67 1 11 2 -5 7 67 11 6 1 -5 7 -47 11 1 11 0 7 -47 524
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -29 mod 625 ≡ 596n b q r t1 t2 t3 625 88556 0 625 0 1 0 88556 625 141 431 1 0 1 625 431 1 194 0 1 -1 431 194 2 43 1 -1 3 194 43 4 22 -1 3 -13 43 22 1 21 3 -13 16 22 21 1 1 -13 16 -29 21 1 21 0 16 -29 625
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 23 mod 169 ≡ 23n b q r t1 t2 t3 169 327500 0 169 0 1 0 327500 169 1937 147 1 0 1 169 147 1 22 0 1 -1 147 22 6 15 1 -1 7 22 15 1 7 -1 7 -8 15 7 2 1 7 -8 23 7 1 7 0 -8 23 -169
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
= (172 × 105625 × 477 +
622 × 88556 × 596 +
35 × 327500 × 23) mod 55347500
= 26291872 (mod 55347500)
So our answer is 26291872 (mod 55347500).
Verification
So we found that x ≡ 26291872
If this is correct, then the following statements (i.e. the original equations) are true:
525x (mod 524) ≡ 172 (mod 524)
504x (mod 625) ≡ 363 (mod 625)
36x (mod 169) ≡ 246 (mod 169)
Let's see whether that's indeed the case if we use x ≡ 26291872.