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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
82119046101-4
19061371-413
61785-413-108
751213-108121
5221-108121-350
2120121-350821
So our multiplicative inverse is -350 mod 821 ≡ 471
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
2578700257010
870257399101
2579925901-2
99591401-23
5940119-23-5
4019223-513
19291-513-122
212013-122257
So our multiplicative inverse is -122 mod 257 ≡ 135
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
71669071010
66971930101
713021101-2
3011281-25
11813-25-7
83225-719
3211-719-26
212019-2671
So our multiplicative inverse is -26 mod 71 ≡ 45
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 ≡ 58 × 190-1 (mod 821) ≡ 58 × 471 (mod 821) ≡ 225 (mod 821)
x ≡ 730 × 870-1 (mod 257) ≡ 730 × 135 (mod 257) ≡ 119 (mod 257)
x ≡ 902 × 669-1 (mod 71) ≡ 902 × 45 (mod 71) ≡ 49 (mod 71)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 821 × 257 × 71 = 14980787
  2. We calculate the numbers M1 to M3
    M1=M/m1=14980787/821=18247,   M2=M/m2=14980787/257=58291,   M3=M/m3=14980787/71=210997
  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
    821182470821010
    1824782122185101
    82118548101-4
    185812231-49
    8123312-49-31
    23121119-3140
    121111-3140-71
    11111040-71821
    So our multiplicative inverse is -71 mod 821 ≡ 750
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    257582910257010
    58291257226209101
    25720914801-1
    209484171-15
    4817214-15-11
    1714135-1116
    14342-1116-75
    321116-7591
    2120-7591-257
    So our multiplicative inverse is 91 mod 257 ≡ 91
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    71210997071010
    21099771297156101
    715611501-1
    56153111-14
    151114-14-5
    114234-514
    4311-514-19
    313014-1971
    So our multiplicative inverse is -19 mod 71 ≡ 52
    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
    =  (225 × 18247 × 750 +
       119 × 58291 × 91 +
       49 × 210997 × 52)   mod 14980787
    = 8472124 (mod 14980787)


    So our answer is 8472124 (mod 14980787).


Verification

So we found that x ≡ 8472124
If this is correct, then the following statements (i.e. the original equations) are true:
190x (mod 821) ≡ 58 (mod 821)
870x (mod 257) ≡ 730 (mod 257)
669x (mod 71) ≡ 902 (mod 71)

Let's see whether that's indeed the case if we use x ≡ 8472124.