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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!

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
872657121501-1
6572153121-14
215121711-14-69
1211114-6973
111110-6973-872
So our multiplicative inverse is 73 mod 872 ≡ 73
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
7819383701-8
93372191-817
3719118-817-25
19181117-2542
181180-2542-781
So our multiplicative inverse is 42 mod 781 ≡ 42
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
577390118701-1
3901872161-13
187161111-13-34
1611153-3437
11521-3437-108
515037-108577
So our multiplicative inverse is -108 mod 577 ≡ 469
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 ≡ 890 × 657-1 (mod 872) ≡ 890 × 73 (mod 872) ≡ 442 (mod 872)
x ≡ 194 × 93-1 (mod 781) ≡ 194 × 42 (mod 781) ≡ 338 (mod 781)
x ≡ 240 × 390-1 (mod 577) ≡ 240 × 469 (mod 577) ≡ 45 (mod 577)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 872 × 781 × 577 = 392955464
  2. We calculate the numbers M1 to M3
    M1=M/m1=392955464/872=450637,   M2=M/m2=392955464/781=503144,   M3=M/m3=392955464/577=681032
  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
    8724506370872010
    450637872516685101
    872685118701-1
    68518731241-14
    187124163-14-5
    124631614-59
    636112-59-14
    6123019-14429
    2120-14429-872
    So our multiplicative inverse is 429 mod 872 ≡ 429
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    7815031440781010
    503144781644180101
    78118046101-4
    180612581-49
    615813-49-13
    5831919-13256
    3130-13256-781
    So our multiplicative inverse is 256 mod 781 ≡ 256
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    5776810320577010
    6810325771180172101
    57717236101-3
    172612501-37
    6150111-37-10
    5011467-1047
    11615-1047-57
    651147-57104
    5150-57104-577
    So our multiplicative inverse is 104 mod 577 ≡ 104
    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
    =  (442 × 450637 × 429 +
       338 × 503144 × 256 +
       45 × 681032 × 104)   mod 392955464
    = 139124554 (mod 392955464)


    So our answer is 139124554 (mod 392955464).


Verification

So we found that x ≡ 139124554
If this is correct, then the following statements (i.e. the original equations) are true:
657x (mod 872) ≡ 890 (mod 872)
93x (mod 781) ≡ 194 (mod 781)
390x (mod 577) ≡ 240 (mod 577)

Let's see whether that's indeed the case if we use x ≡ 139124554.