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
1098012901-1
80292221-13
292217-13-4
227313-415
7170-415-109
So our multiplicative inverse is 15 mod 109 ≡ 15
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
2773110277010
311277134101
277348501-8
345641-849
5411-849-57
414049-57277
So our multiplicative inverse is -57 mod 277 ≡ 220
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
809324216101-2
324161221-25
1612801-25-402
21205-402809
So our multiplicative inverse is -402 mod 809 ≡ 407
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 ≡ 798 × 80-1 (mod 109) ≡ 798 × 15 (mod 109) ≡ 89 (mod 109)
x ≡ 172 × 311-1 (mod 277) ≡ 172 × 220 (mod 277) ≡ 168 (mod 277)
x ≡ 974 × 324-1 (mod 809) ≡ 974 × 407 (mod 809) ≡ 8 (mod 809)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 109 × 277 × 809 = 24426137
  2. We calculate the numbers M1 to M3
    M1=M/m1=24426137/109=224093,   M2=M/m2=24426137/277=88181,   M3=M/m3=24426137/809=30193
  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
    1092240930109010
    224093109205598101
    1099811101-1
    98118101-19
    111011-19-10
    1011009-10109
    So our multiplicative inverse is -10 mod 109 ≡ 99
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    277881810277010
    8818127731895101
    2779528701-2
    9587181-23
    878107-23-32
    87113-3235
    7170-3235-277
    So our multiplicative inverse is 35 mod 277 ≡ 35
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    809301930809010
    3019380937260101
    80926032901-3
    260298281-325
    292811-325-28
    28128025-28809
    So our multiplicative inverse is -28 mod 809 ≡ 781
    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
    =  (89 × 224093 × 99 +
       168 × 88181 × 35 +
       8 × 30193 × 781)   mod 24426137
    = 19184634 (mod 24426137)


    So our answer is 19184634 (mod 24426137).


Verification

So we found that x ≡ 19184634
If this is correct, then the following statements (i.e. the original equations) are true:
80x (mod 109) ≡ 798 (mod 109)
311x (mod 277) ≡ 172 (mod 277)
324x (mod 809) ≡ 974 (mod 809)

Let's see whether that's indeed the case if we use x ≡ 19184634.