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Welcome to ChineseRemainderTheorem.com!

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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!

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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
88682715901-1
827591411-115
591590-115-886
So our multiplicative inverse is 15 mod 886 ≡ 15
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
683454122901-1
45422912251-12
22922514-12-3
22545612-3170
4140-3170-683
So our multiplicative inverse is 170 mod 683 ≡ 170
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
6679850667010
9856671318101
66731823101-2
318311081-221
31837-221-65
871121-6586
7170-6586-667
So our multiplicative inverse is 86 mod 667 ≡ 86
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 ≡ 391 × 827-1 (mod 886) ≡ 391 × 15 (mod 886) ≡ 549 (mod 886)
x ≡ 891 × 454-1 (mod 683) ≡ 891 × 170 (mod 683) ≡ 527 (mod 683)
x ≡ 343 × 985-1 (mod 667) ≡ 343 × 86 (mod 667) ≡ 150 (mod 667)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 886 × 683 × 667 = 403627046
  2. We calculate the numbers M1 to M3
    M1=M/m1=403627046/886=455561,   M2=M/m2=403627046/683=590962,   M3=M/m3=403627046/667=605138
  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
    8864555610886010
    455561886514157101
    886157510101-5
    1571011561-56
    10156145-56-11
    56451116-1117
    451141-1117-79
    11111017-79886
    So our multiplicative inverse is -79 mod 886 ≡ 807
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    6835909620683010
    590962683865167101
    68316741501-4
    167151121-445
    15271-445-319
    212045-319683
    So our multiplicative inverse is -319 mod 683 ≡ 364
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    6676051380667010
    605138667907169101
    667169316001-3
    169160191-34
    1609177-34-71
    97124-7175
    7231-7175-296
    212075-296667
    So our multiplicative inverse is -296 mod 667 ≡ 371
    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
    =  (549 × 455561 × 807 +
       527 × 590962 × 364 +
       150 × 605138 × 371)   mod 403627046
    = 138332615 (mod 403627046)


    So our answer is 138332615 (mod 403627046).


Verification

So we found that x ≡ 138332615
If this is correct, then the following statements (i.e. the original equations) are true:
827x (mod 886) ≡ 391 (mod 886)
454x (mod 683) ≡ 891 (mod 683)
985x (mod 667) ≡ 343 (mod 667)

Let's see whether that's indeed the case if we use x ≡ 138332615.