Bootstrap
  C.R.T. .com
It doesn't have to be difficult if someone just explains it right.

Welcome to ChineseRemainderTheorem.com!

×

Modal Header

Some text in the Modal Body

Some other text...

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
73941073010
941731265101
73651801-1
658811-19
8180-19-73
So our multiplicative inverse is 9 mod 73 ≡ 9
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
2279250227010
925227417101
2271713601-13
176251-1327
6511-1327-40
515027-40227
So our multiplicative inverse is -40 mod 227 ≡ 187
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
6539640653010
9646531311101
65331123101-2
311311011-221
311310-221-653
So our multiplicative inverse is 21 mod 653 ≡ 21
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 ≡ 952 × 941-1 (mod 73) ≡ 952 × 9 (mod 73) ≡ 27 (mod 73)
x ≡ 923 × 925-1 (mod 227) ≡ 923 × 187 (mod 227) ≡ 81 (mod 227)
x ≡ 853 × 964-1 (mod 653) ≡ 853 × 21 (mod 653) ≡ 282 (mod 653)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 73 × 227 × 653 = 10820863
  2. We calculate the numbers M1 to M3
    M1=M/m1=10820863/73=148231,   M2=M/m2=10820863/227=47669,   M3=M/m3=10820863/653=16571
  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
    73148231073010
    14823173203041101
    734113201-1
    4132191-12
    32935-12-7
    95142-79
    5411-79-16
    41409-1673
    So our multiplicative inverse is -16 mod 73 ≡ 57
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    227476690227010
    47669227209226101
    2272261101-1
    226122601-1227
    So our multiplicative inverse is -1 mod 227 ≡ 226
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    653165710653010
    1657165325246101
    653246216101-2
    2461611851-23
    16185176-23-5
    8576193-58
    76984-58-69
    94218-69146
    4140-69146-653
    So our multiplicative inverse is 146 mod 653 ≡ 146
    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
    =  (27 × 148231 × 57 +
       81 × 47669 × 226 +
       282 × 16571 × 146)   mod 10820863
    = 8395903 (mod 10820863)


    So our answer is 8395903 (mod 10820863).


Verification

So we found that x ≡ 8395903
If this is correct, then the following statements (i.e. the original equations) are true:
941x (mod 73) ≡ 952 (mod 73)
925x (mod 227) ≡ 923 (mod 227)
964x (mod 653) ≡ 853 (mod 653)

Let's see whether that's indeed the case if we use x ≡ 8395903.