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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
2415100241010
510241228101
2412881701-8
28171111-89
171116-89-17
116159-1726
6511-1726-43
515026-43241
So our multiplicative inverse is -43 mod 241 ≡ 198
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
919725119401-1
72519431431-14
194143151-14-5
143512414-514
5141110-514-19
41104114-1990
101100-1990-919
So our multiplicative inverse is 90 mod 919 ≡ 90
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
4699540469010
954469216101
4691629501-29
165311-2988
5150-2988-469
So our multiplicative inverse is 88 mod 469 ≡ 88
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 ≡ 853 × 510-1 (mod 241) ≡ 853 × 198 (mod 241) ≡ 194 (mod 241)
x ≡ 713 × 725-1 (mod 919) ≡ 713 × 90 (mod 919) ≡ 759 (mod 919)
x ≡ 259 × 954-1 (mod 469) ≡ 259 × 88 (mod 469) ≡ 280 (mod 469)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 241 × 919 × 469 = 103873651
  2. We calculate the numbers M1 to M3
    M1=M/m1=103873651/241=431011,   M2=M/m2=103873651/919=113029,   M3=M/m3=103873651/469=221479
  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
    2414310110241010
    4310112411788103101
    24110323501-2
    103352331-25
    353312-25-7
    3321615-7117
    2120-7117-241
    So our multiplicative inverse is 117 mod 241 ≡ 117
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    9191130290919010
    113029919122911101
    9199111801-1
    911811371-1114
    8711-1114-115
    7170114-115919
    So our multiplicative inverse is -115 mod 919 ≡ 804
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    4692214790469010
    221479469472111101
    46911142501-4
    111254111-417
    251123-417-38
    1133217-38131
    3211-38131-169
    2120131-169469
    So our multiplicative inverse is -169 mod 469 ≡ 300
    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
    =  (194 × 431011 × 117 +
       759 × 113029 × 804 +
       280 × 221479 × 300)   mod 103873651
    = 32077535 (mod 103873651)


    So our answer is 32077535 (mod 103873651).


Verification

So we found that x ≡ 32077535
If this is correct, then the following statements (i.e. the original equations) are true:
510x (mod 241) ≡ 853 (mod 241)
725x (mod 919) ≡ 713 (mod 919)
954x (mod 469) ≡ 259 (mod 469)

Let's see whether that's indeed the case if we use x ≡ 32077535.