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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
971144610701-6
1441071371-67
10737233-67-20
3733147-2027
33481-2027-236
414027-236971
So our multiplicative inverse is -236 mod 971 ≡ 735
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
71327071010
32771443101
714312801-1
43281151-12
2815113-12-3
1513122-35
13261-35-33
21205-3371
So our multiplicative inverse is -33 mod 71 ≡ 38
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
929582134701-1
58234712351-12
3472351112-12-3
2351122112-38
11211102-38-83
112518-83423
2120-83423-929
So our multiplicative inverse is 423 mod 929 ≡ 423
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 ≡ 94 × 144-1 (mod 971) ≡ 94 × 735 (mod 971) ≡ 149 (mod 971)
x ≡ 991 × 327-1 (mod 71) ≡ 991 × 38 (mod 71) ≡ 28 (mod 71)
x ≡ 49 × 582-1 (mod 929) ≡ 49 × 423 (mod 929) ≡ 289 (mod 929)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 971 × 71 × 929 = 64046189
  2. We calculate the numbers M1 to M3
    M1=M/m1=64046189/971=65959,   M2=M/m2=64046189/71=902059,   M3=M/m3=64046189/929=68941
  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
    971659590971010
    6595997167902101
    97190216901-1
    902691351-114
    695134-114-183
    541114-183197
    4140-183197-971
    So our multiplicative inverse is 197 mod 971 ≡ 197
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    71902059071010
    90205971127054101
    71417301-17
    43111-1718
    3130-1718-71
    So our multiplicative inverse is 18 mod 71 ≡ 18
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    929689410929010
    6894192974195101
    929195414901-4
    1951491461-45
    14946311-45-19
    4611425-1981
    11251-1981-424
    212081-424929
    So our multiplicative inverse is -424 mod 929 ≡ 505
    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
    =  (149 × 65959 × 197 +
       28 × 902059 × 18 +
       289 × 68941 × 505)   mod 64046189
    = 27365842 (mod 64046189)


    So our answer is 27365842 (mod 64046189).


Verification

So we found that x ≡ 27365842
If this is correct, then the following statements (i.e. the original equations) are true:
144x (mod 971) ≡ 94 (mod 971)
327x (mod 71) ≡ 991 (mod 71)
582x (mod 929) ≡ 49 (mod 929)

Let's see whether that's indeed the case if we use x ≡ 27365842.