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
4497780449010
7784491329101
449329112001-1
3291202891-13
12089131-13-4
89312273-411
312714-411-15
2746311-15101
4311-15101-116
3130101-116449
So our multiplicative inverse is -116 mod 449 ≡ 333
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
32714024701-2
140472461-25
474611-25-7
4614605-7327
So our multiplicative inverse is -7 mod 327 ≡ 320
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
4336370433010
6374331204101
43320422501-2
20425841-217
25461-217-104
414017-104433
So our multiplicative inverse is -104 mod 433 ≡ 329
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 ≡ 851 × 778-1 (mod 449) ≡ 851 × 333 (mod 449) ≡ 64 (mod 449)
x ≡ 207 × 140-1 (mod 327) ≡ 207 × 320 (mod 327) ≡ 186 (mod 327)
x ≡ 192 × 637-1 (mod 433) ≡ 192 × 329 (mod 433) ≡ 383 (mod 433)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 449 × 327 × 433 = 63574359
  2. We calculate the numbers M1 to M3
    M1=M/m1=63574359/449=141591,   M2=M/m2=63574359/327=194417,   M3=M/m3=63574359/433=146823
  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
    4491415910449010
    141591449315156101
    449156213701-2
    1561371191-23
    1371974-23-23
    194433-2395
    4311-2395-118
    313095-118449
    So our multiplicative inverse is -118 mod 449 ≡ 331
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    3271944170327010
    194417327594179101
    327179114801-1
    1791481311-12
    14831424-12-9
    3124172-911
    24733-911-42
    732111-4295
    3130-4295-327
    So our multiplicative inverse is 95 mod 327 ≡ 95
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    4331468230433010
    14682343333936101
    4333612101-12
    3613601-12433
    So our multiplicative inverse is -12 mod 433 ≡ 421
    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
    =  (64 × 141591 × 331 +
       186 × 194417 × 95 +
       383 × 146823 × 421)   mod 63574359
    = 38321316 (mod 63574359)


    So our answer is 38321316 (mod 63574359).


Verification

So we found that x ≡ 38321316
If this is correct, then the following statements (i.e. the original equations) are true:
778x (mod 449) ≡ 851 (mod 449)
140x (mod 327) ≡ 207 (mod 327)
637x (mod 433) ≡ 192 (mod 433)

Let's see whether that's indeed the case if we use x ≡ 38321316.