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
5245250524010
52552411101
5241524001-524
So our multiplicative inverse is 1 mod 524 ≡ 1
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
625504112101-1
5041214201-15
1212061-15-31
2012005-31625
So our multiplicative inverse is -31 mod 625 ≡ 594
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
1693642501-4
36251111-45
251123-45-14
113325-1447
3211-1447-61
212047-61169
So our multiplicative inverse is -61 mod 169 ≡ 108
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 ≡ 172 × 525-1 (mod 524) ≡ 172 × 1 (mod 524) ≡ 172 (mod 524)
x ≡ 363 × 504-1 (mod 625) ≡ 363 × 594 (mod 625) ≡ 622 (mod 625)
x ≡ 246 × 36-1 (mod 169) ≡ 246 × 108 (mod 169) ≡ 35 (mod 169)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 524 × 625 × 169 = 55347500
  2. We calculate the numbers M1 to M3
    M1=M/m1=55347500/524=105625,   M2=M/m2=55347500/625=88556,   M3=M/m3=55347500/169=327500
  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
    5241056250524010
    105625524201301101
    524301122301-1
    3012231781-12
    22378267-12-5
    78671112-57
    671161-57-47
    1111107-47524
    So our multiplicative inverse is -47 mod 524 ≡ 477
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    625885560625010
    88556625141431101
    625431119401-1
    4311942431-13
    19443422-13-13
    43221213-1316
    222111-1316-29
    21121016-29625
    So our multiplicative inverse is -29 mod 625 ≡ 596
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    1693275000169010
    3275001691937147101
    16914712201-1
    147226151-17
    221517-17-8
    157217-823
    7170-823-169
    So our multiplicative inverse is 23 mod 169 ≡ 23
    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
    =  (172 × 105625 × 477 +
       622 × 88556 × 596 +
       35 × 327500 × 23)   mod 55347500
    = 26291872 (mod 55347500)


    So our answer is 26291872 (mod 55347500).


Verification

So we found that x ≡ 26291872
If this is correct, then the following statements (i.e. the original equations) are true:
525x (mod 524) ≡ 172 (mod 524)
504x (mod 625) ≡ 363 (mod 625)
36x (mod 169) ≡ 246 (mod 169)

Let's see whether that's indeed the case if we use x ≡ 26291872.