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
959632132701-1
63232713051-12
327305122-12-3
3052213192-341
221913-341-44
1936141-44305
3130-44305-959
So our multiplicative inverse is 305 mod 959 ≡ 305
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
1215630121010
563121479101
1217914201-1
79421371-12
423715-12-3
375722-323
5221-323-49
212023-49121
So our multiplicative inverse is -49 mod 121 ≡ 72
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
1698560169010
856169511101
1691115401-15
114231-1531
4311-1531-46
313031-46169
So our multiplicative inverse is -46 mod 169 ≡ 123
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 ≡ 80 × 632-1 (mod 959) ≡ 80 × 305 (mod 959) ≡ 425 (mod 959)
x ≡ 212 × 563-1 (mod 121) ≡ 212 × 72 (mod 121) ≡ 18 (mod 121)
x ≡ 49 × 856-1 (mod 169) ≡ 49 × 123 (mod 169) ≡ 112 (mod 169)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 959 × 121 × 169 = 19610591
  2. We calculate the numbers M1 to M3
    M1=M/m1=19610591/959=20449,   M2=M/m2=19610591/121=162071,   M3=M/m3=19610591/169=116039
  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
    959204490959010
    2044995921310101
    95931032901-3
    3102910201-331
    292019-331-34
    2092231-3499
    9241-3499-430
    212099-430959
    So our multiplicative inverse is -430 mod 959 ≡ 529
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    1211620710121010
    162071121133952101
    1215221701-2
    5217311-27
    171170-27-121
    So our multiplicative inverse is 7 mod 121 ≡ 7
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    1691160390169010
    116039169686105101
    16910516401-1
    105641411-12
    6441123-12-3
    41231182-35
    231815-35-8
    185335-829
    5312-829-37
    321129-3766
    2120-3766-169
    So our multiplicative inverse is 66 mod 169 ≡ 66
    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
    =  (425 × 20449 × 529 +
       18 × 162071 × 7 +
       112 × 116039 × 66)   mod 19610591
    = 4272770 (mod 19610591)


    So our answer is 4272770 (mod 19610591).


Verification

So we found that x ≡ 4272770
If this is correct, then the following statements (i.e. the original equations) are true:
632x (mod 959) ≡ 80 (mod 959)
563x (mod 121) ≡ 212 (mod 121)
856x (mod 169) ≡ 49 (mod 169)

Let's see whether that's indeed the case if we use x ≡ 4272770.