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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.

This fills all textboxes with random numbers. If you fill in random numbers yourself, it is very likely that those numbers do not have a solution. To avoid disappointment, use this button instead! It only uses random numbers that do have a solution.

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.

Do you want to use more equations? Go ahead and use this button. It adds another row that you can fill in. Not sure what numbers to put in this newly added row? Use the random numbers button again!

Do you have too many rows? Use one of these buttons to remove a row. You can always add a row again using the yellow "Add a row" button.

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?


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:

nbqr t1t2t3
5129490512010
9495121437101
51243717501-1
437755621-16
7562113-16-7
62134106-734
131013-734-41
1033134-41157
3130-41157-512
So our multiplicative inverse is 157 mod 512 ≡ 157
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
647440120701-1
4402072261-13
20726725-13-22
2625113-2225
251250-2225-647
So our multiplicative inverse is 25 mod 647 ≡ 25
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
6532724501-24
275521-24121
5221-24121-266
2120121-266653
So our multiplicative inverse is -266 mod 653 ≡ 387
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 ≡ 866 × 949-1 (mod 512) ≡ 866 × 157 (mod 512) ≡ 282 (mod 512)
x ≡ 405 × 440-1 (mod 647) ≡ 405 × 25 (mod 647) ≡ 420 (mod 647)
x ≡ 778 × 27-1 (mod 653) ≡ 778 × 387 (mod 653) ≡ 53 (mod 653)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 512 × 647 × 653 = 216315392
  2. We calculate the numbers M1 to M3
    M1=M/m1=216315392/512=422491,   M2=M/m2=216315392/647=334336,   M3=M/m3=216315392/653=331264
  3. 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:

    nbqr t1t2t3
    5124224910512010
    42249151282591101
    5129155701-5
    91571341-56
    5734123-56-11
    34231116-1117
    231121-1117-45
    11111017-45512
    So our multiplicative inverse is -45 mod 512 ≡ 467
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    6473343360647010
    334336647516484101
    647484116301-1
    48416321581-13
    16315815-13-4
    15853133-4127
    5312-4127-131
    3211127-131258
    2120-131258-647
    So our multiplicative inverse is 258 mod 647 ≡ 258
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    6533312640653010
    331264653507193101
    65319337401-3
    193742451-37
    7445129-37-10
    45291167-1017
    2916113-1017-27
    16131317-2744
    13341-2744-203
    313044-203653
    So our multiplicative inverse is -203 mod 653 ≡ 450
    Source: ExtendedEuclideanAlgorithm.com
  4. 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
    =  (282 × 422491 × 467 +
       420 × 334336 × 258 +
       53 × 331264 × 450)   mod 216315392
    = 47429402 (mod 216315392)


    So our answer is 47429402 (mod 216315392).


Verification

So we found that x ≡ 47429402
If this is correct, then the following statements (i.e. the original equations) are true:
949x (mod 512) ≡ 866 (mod 512)
440x (mod 647) ≡ 405 (mod 647)
27x (mod 653) ≡ 778 (mod 653)

Let's see whether that's indeed the case if we use x ≡ 47429402.