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 button is similar to the "Clear everything" button, but only clears the left column.
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 |
|---|---|---|---|---|---|---|
| 419 | 106 | 3 | 101 | 0 | 1 | -3 |
| 106 | 101 | 1 | 5 | 1 | -3 | 4 |
| 101 | 5 | 20 | 1 | -3 | 4 | -83 |
| 5 | 1 | 5 | 0 | 4 | -83 | 419 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 137 | 383 | 0 | 137 | 0 | 1 | 0 |
| 383 | 137 | 2 | 109 | 1 | 0 | 1 |
| 137 | 109 | 1 | 28 | 0 | 1 | -1 |
| 109 | 28 | 3 | 25 | 1 | -1 | 4 |
| 28 | 25 | 1 | 3 | -1 | 4 | -5 |
| 25 | 3 | 8 | 1 | 4 | -5 | 44 |
| 3 | 1 | 3 | 0 | -5 | 44 | -137 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 601 | 50 | 12 | 1 | 0 | 1 | -12 |
| 50 | 1 | 50 | 0 | 1 | -12 | 601 |
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 ≡ 115 × 106-1 (mod 419) ≡ 115 × 336 (mod 419) ≡ 92 (mod 419)x ≡ 179 × 383-1 (mod 137) ≡ 179 × 44 (mod 137) ≡ 67 (mod 137)x ≡ 928 × 50-1 (mod 601) ≡ 928 × 589 (mod 601) ≡ 283 (mod 601)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 92 (mod 419)
x ≡ 67 (mod 137)
x ≡ 283 (mod 601)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 419 × 137 × 601 = 34499203 - We calculate the numbers M1 to M3
M1=M/m1=34499203/419=82337,M2=M/m2=34499203/137=251819,M3=M/m3=34499203/601=57403 - 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 120 mod 419 ≡ 120n b q r t1 t2 t3 419 82337 0 419 0 1 0 82337 419 196 213 1 0 1 419 213 1 206 0 1 -1 213 206 1 7 1 -1 2 206 7 29 3 -1 2 -59 7 3 2 1 2 -59 120 3 1 3 0 -59 120 -419
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -21 mod 137 ≡ 116n b q r t1 t2 t3 137 251819 0 137 0 1 0 251819 137 1838 13 1 0 1 137 13 10 7 0 1 -10 13 7 1 6 1 -10 11 7 6 1 1 -10 11 -21 6 1 6 0 11 -21 137
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -80 mod 601 ≡ 521n b q r t1 t2 t3 601 57403 0 601 0 1 0 57403 601 95 308 1 0 1 601 308 1 293 0 1 -1 308 293 1 15 1 -1 2 293 15 19 8 -1 2 -39 15 8 1 7 2 -39 41 8 7 1 1 -39 41 -80 7 1 7 0 41 -80 601
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
= (92 × 82337 × 120 +
67 × 251819 × 116 +
283 × 57403 × 521) mod 34499203
= 14069693 (mod 34499203)
So our answer is 14069693 (mod 34499203).
Verification
So we found that x ≡ 14069693
If this is correct, then the following statements (i.e. the original equations) are true:
106x (mod 419) ≡ 115 (mod 419)
383x (mod 137) ≡ 179 (mod 137)
50x (mod 601) ≡ 928 (mod 601)
Let's see whether that's indeed the case if we use x ≡ 14069693.