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Welcome to ChineseRemainderTheorem.com!

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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
29923316601-1
233663351-14
6635131-14-5
3531144-59
31473-59-68
43119-6877
3130-6877-299
So our multiplicative inverse is 77 mod 299 ≡ 77
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
934373218801-2
37318811851-23
18818513-23-5
18536123-5308
3211-5308-313
2120308-313934
So our multiplicative inverse is -313 mod 934 ≡ 621
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
97752184101-18
52411111-1819
411138-1819-75
1181319-7594
8322-7594-263
321194-263357
2120-263357-977
So our multiplicative inverse is 357 mod 977 ≡ 357
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 ≡ 288 × 233-1 (mod 299) ≡ 288 × 77 (mod 299) ≡ 50 (mod 299)
x ≡ 770 × 373-1 (mod 934) ≡ 770 × 621 (mod 934) ≡ 896 (mod 934)
x ≡ 734 × 52-1 (mod 977) ≡ 734 × 357 (mod 977) ≡ 202 (mod 977)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 299 × 934 × 977 = 272842882
  2. We calculate the numbers M1 to M3
    M1=M/m1=272842882/299=912518,   M2=M/m2=272842882/934=292123,   M3=M/m3=272842882/977=279266
  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
    2999125180299010
    9125182993051269101
    29926913001-1
    269308291-19
    302911-19-10
    2912909-10299
    So our multiplicative inverse is -10 mod 299 ≡ 289
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    9342921230934010
    292123934312715101
    934715121901-1
    7152193581-14
    21958345-14-13
    58451134-1317
    451336-1317-64
    1362117-64145
    6160-64145-934
    So our multiplicative inverse is 145 mod 934 ≡ 145
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    9772792660977010
    279266977285821101
    977821115601-1
    8211565411-16
    15641333-16-19
    4133186-1925
    33841-1925-119
    818025-119977
    So our multiplicative inverse is -119 mod 977 ≡ 858
    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
    =  (50 × 912518 × 289 +
       896 × 292123 × 145 +
       202 × 279266 × 858)   mod 272842882
    = 224962268 (mod 272842882)


    So our answer is 224962268 (mod 272842882).


Verification

So we found that x ≡ 224962268
If this is correct, then the following statements (i.e. the original equations) are true:
233x (mod 299) ≡ 288 (mod 299)
373x (mod 934) ≡ 770 (mod 934)
52x (mod 977) ≡ 734 (mod 977)

Let's see whether that's indeed the case if we use x ≡ 224962268.