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 |
|---|---|---|---|---|---|---|
| 971 | 640 | 1 | 331 | 0 | 1 | -1 |
| 640 | 331 | 1 | 309 | 1 | -1 | 2 |
| 331 | 309 | 1 | 22 | -1 | 2 | -3 |
| 309 | 22 | 14 | 1 | 2 | -3 | 44 |
| 22 | 1 | 22 | 0 | -3 | 44 | -971 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 929 | 516 | 1 | 413 | 0 | 1 | -1 |
| 516 | 413 | 1 | 103 | 1 | -1 | 2 |
| 413 | 103 | 4 | 1 | -1 | 2 | -9 |
| 103 | 1 | 103 | 0 | 2 | -9 | 929 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 653 | 988 | 0 | 653 | 0 | 1 | 0 |
| 988 | 653 | 1 | 335 | 1 | 0 | 1 |
| 653 | 335 | 1 | 318 | 0 | 1 | -1 |
| 335 | 318 | 1 | 17 | 1 | -1 | 2 |
| 318 | 17 | 18 | 12 | -1 | 2 | -37 |
| 17 | 12 | 1 | 5 | 2 | -37 | 39 |
| 12 | 5 | 2 | 2 | -37 | 39 | -115 |
| 5 | 2 | 2 | 1 | 39 | -115 | 269 |
| 2 | 1 | 2 | 0 | -115 | 269 | -653 |
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 ≡ 220 × 640-1 (mod 971) ≡ 220 × 44 (mod 971) ≡ 941 (mod 971)x ≡ 624 × 516-1 (mod 929) ≡ 624 × 920 (mod 929) ≡ 887 (mod 929)x ≡ 186 × 988-1 (mod 653) ≡ 186 × 269 (mod 653) ≡ 406 (mod 653)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 941 (mod 971)
x ≡ 887 (mod 929)
x ≡ 406 (mod 653)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 971 × 929 × 653 = 589044527 - We calculate the numbers M1 to M3
M1=M/m1=589044527/971=606637,M2=M/m2=589044527/929=634063,M3=M/m3=589044527/653=902059 - 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 -102 mod 971 ≡ 869n b q r t1 t2 t3 971 606637 0 971 0 1 0 606637 971 624 733 1 0 1 971 733 1 238 0 1 -1 733 238 3 19 1 -1 4 238 19 12 10 -1 4 -49 19 10 1 9 4 -49 53 10 9 1 1 -49 53 -102 9 1 9 0 53 -102 971
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 136 mod 929 ≡ 136n b q r t1 t2 t3 929 634063 0 929 0 1 0 634063 929 682 485 1 0 1 929 485 1 444 0 1 -1 485 444 1 41 1 -1 2 444 41 10 34 -1 2 -21 41 34 1 7 2 -21 23 34 7 4 6 -21 23 -113 7 6 1 1 23 -113 136 6 1 6 0 -113 136 -929
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -27 mod 653 ≡ 626n b q r t1 t2 t3 653 902059 0 653 0 1 0 902059 653 1381 266 1 0 1 653 266 2 121 0 1 -2 266 121 2 24 1 -2 5 121 24 5 1 -2 5 -27 24 1 24 0 5 -27 653
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
= (941 × 606637 × 869 +
887 × 634063 × 136 +
406 × 902059 × 626) mod 589044527
= 127061146 (mod 589044527)
So our answer is 127061146 (mod 589044527).
Verification
So we found that x ≡ 127061146
If this is correct, then the following statements (i.e. the original equations) are true:
640x (mod 971) ≡ 220 (mod 971)
516x (mod 929) ≡ 624 (mod 929)
988x (mod 653) ≡ 186 (mod 653)
Let's see whether that's indeed the case if we use x ≡ 127061146.