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 |
|---|---|---|---|---|---|---|
| 674 | 891 | 0 | 674 | 0 | 1 | 0 |
| 891 | 674 | 1 | 217 | 1 | 0 | 1 |
| 674 | 217 | 3 | 23 | 0 | 1 | -3 |
| 217 | 23 | 9 | 10 | 1 | -3 | 28 |
| 23 | 10 | 2 | 3 | -3 | 28 | -59 |
| 10 | 3 | 3 | 1 | 28 | -59 | 205 |
| 3 | 1 | 3 | 0 | -59 | 205 | -674 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 641 | 403 | 1 | 238 | 0 | 1 | -1 |
| 403 | 238 | 1 | 165 | 1 | -1 | 2 |
| 238 | 165 | 1 | 73 | -1 | 2 | -3 |
| 165 | 73 | 2 | 19 | 2 | -3 | 8 |
| 73 | 19 | 3 | 16 | -3 | 8 | -27 |
| 19 | 16 | 1 | 3 | 8 | -27 | 35 |
| 16 | 3 | 5 | 1 | -27 | 35 | -202 |
| 3 | 1 | 3 | 0 | 35 | -202 | 641 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 943 | 387 | 2 | 169 | 0 | 1 | -2 |
| 387 | 169 | 2 | 49 | 1 | -2 | 5 |
| 169 | 49 | 3 | 22 | -2 | 5 | -17 |
| 49 | 22 | 2 | 5 | 5 | -17 | 39 |
| 22 | 5 | 4 | 2 | -17 | 39 | -173 |
| 5 | 2 | 2 | 1 | 39 | -173 | 385 |
| 2 | 1 | 2 | 0 | -173 | 385 | -943 |
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 ≡ 636 × 891-1 (mod 674) ≡ 636 × 205 (mod 674) ≡ 298 (mod 674)x ≡ 899 × 403-1 (mod 641) ≡ 899 × 439 (mod 641) ≡ 446 (mod 641)x ≡ 674 × 387-1 (mod 943) ≡ 674 × 385 (mod 943) ≡ 165 (mod 943)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 298 (mod 674)
x ≡ 446 (mod 641)
x ≡ 165 (mod 943)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 674 × 641 × 943 = 407408062 - We calculate the numbers M1 to M3
M1=M/m1=407408062/674=604463,M2=M/m2=407408062/641=635582,M3=M/m3=407408062/943=432034 - 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 -211 mod 674 ≡ 463n b q r t1 t2 t3 674 604463 0 674 0 1 0 604463 674 896 559 1 0 1 674 559 1 115 0 1 -1 559 115 4 99 1 -1 5 115 99 1 16 -1 5 -6 99 16 6 3 5 -6 41 16 3 5 1 -6 41 -211 3 1 3 0 41 -211 674
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -42 mod 641 ≡ 599n b q r t1 t2 t3 641 635582 0 641 0 1 0 635582 641 991 351 1 0 1 641 351 1 290 0 1 -1 351 290 1 61 1 -1 2 290 61 4 46 -1 2 -9 61 46 1 15 2 -9 11 46 15 3 1 -9 11 -42 15 1 15 0 11 -42 641
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 357 mod 943 ≡ 357n b q r t1 t2 t3 943 432034 0 943 0 1 0 432034 943 458 140 1 0 1 943 140 6 103 0 1 -6 140 103 1 37 1 -6 7 103 37 2 29 -6 7 -20 37 29 1 8 7 -20 27 29 8 3 5 -20 27 -101 8 5 1 3 27 -101 128 5 3 1 2 -101 128 -229 3 2 1 1 128 -229 357 2 1 2 0 -229 357 -943
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
= (298 × 604463 × 463 +
446 × 635582 × 599 +
165 × 432034 × 357) mod 407408062
= 387708014 (mod 407408062)
So our answer is 387708014 (mod 407408062).
Verification
So we found that x ≡ 387708014
If this is correct, then the following statements (i.e. the original equations) are true:
891x (mod 674) ≡ 636 (mod 674)
403x (mod 641) ≡ 899 (mod 641)
387x (mod 943) ≡ 674 (mod 943)
Let's see whether that's indeed the case if we use x ≡ 387708014.