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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!

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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!

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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
4879130487010
9134871426101
48742616101-1
426616601-17
616011-17-8
6016007-8487
So our multiplicative inverse is -8 mod 487 ≡ 479
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
67159317801-1
593787471-18
7847131-18-9
47311168-917
3116115-917-26
16151117-2643
151150-2643-671
So our multiplicative inverse is 43 mod 671 ≡ 43
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
934509142501-1
5094251841-12
4258455-12-11
8451642-11178
5411-11178-189
4140178-189934
So our multiplicative inverse is -189 mod 934 ≡ 745
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 ≡ 947 × 913-1 (mod 487) ≡ 947 × 479 (mod 487) ≡ 216 (mod 487)
x ≡ 980 × 593-1 (mod 671) ≡ 980 × 43 (mod 671) ≡ 538 (mod 671)
x ≡ 549 × 509-1 (mod 934) ≡ 549 × 745 (mod 934) ≡ 847 (mod 934)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 487 × 671 × 934 = 305209718
  2. We calculate the numbers M1 to M3
    M1=M/m1=305209718/487=626714,   M2=M/m2=305209718/671=454858,   M3=M/m3=305209718/934=326777
  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
    4876267140487010
    6267144871286432101
    48743215501-1
    432557471-18
    554718-18-9
    478578-953
    8711-953-62
    717053-62487
    So our multiplicative inverse is -62 mod 487 ≡ 425
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    6714548580671010
    454858671677591101
    67159118001-1
    591807311-18
    8031218-18-17
    31181138-1725
    181315-1725-42
    1352325-42109
    5312-42109-151
    3211109-151260
    2120-151260-671
    So our multiplicative inverse is 260 mod 671 ≡ 260
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    9343267770934010
    326777934349811101
    934811112301-1
    8111236731-17
    12373150-17-8
    73501237-815
    502324-815-38
    2345315-38205
    4311-38205-243
    3130205-243934
    So our multiplicative inverse is -243 mod 934 ≡ 691
    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
    =  (216 × 626714 × 425 +
       538 × 454858 × 260 +
       847 × 326777 × 691)   mod 305209718
    = 183402955 (mod 305209718)


    So our answer is 183402955 (mod 305209718).


Verification

So we found that x ≡ 183402955
If this is correct, then the following statements (i.e. the original equations) are true:
913x (mod 487) ≡ 947 (mod 487)
593x (mod 671) ≡ 980 (mod 671)
509x (mod 934) ≡ 549 (mod 934)

Let's see whether that's indeed the case if we use x ≡ 183402955.