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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
4488950448010
8954481447101
4484471101-1
447144701-1448
So our multiplicative inverse is -1 mod 448 ≡ 447
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
4259070425010
907425257101
4255772601-7
5726251-715
26551-715-82
515015-82425
So our multiplicative inverse is -82 mod 425 ≡ 343
Source: ExtendedEuclideanAlgorithm.com

nbqr t1t2t3
4676810467010
6814671214101
46721423901-2
214395191-211
391921-211-24
19119011-24467
So our multiplicative inverse is -24 mod 467 ≡ 443
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 ≡ 183 × 895-1 (mod 448) ≡ 183 × 447 (mod 448) ≡ 265 (mod 448)
x ≡ 62 × 907-1 (mod 425) ≡ 62 × 343 (mod 425) ≡ 16 (mod 425)
x ≡ 527 × 681-1 (mod 467) ≡ 527 × 443 (mod 467) ≡ 428 (mod 467)


Now the actual calculation

  1. Find the common modulus M
    M = m1 × m2 × ... × mk = 448 × 425 × 467 = 88916800
  2. We calculate the numbers M1 to M3
    M1=M/m1=88916800/448=198475,   M2=M/m2=88916800/425=209216,   M3=M/m3=88916800/467=190400
  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
    4481984750448010
    19847544844311101
    4481140801-40
    118131-4041
    8322-4041-122
    321141-122163
    2120-122163-448
    So our multiplicative inverse is 163 mod 448 ≡ 163
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    4252092160425010
    209216425492116101
    42511637701-3
    116771391-34
    7739138-34-7
    3938114-711
    381380-711-425
    So our multiplicative inverse is 11 mod 425 ≡ 11
    Source: ExtendedEuclideanAlgorithm.com

    nbqr t1t2t3
    4671904000467010
    190400467407331101
    467331113601-1
    3311362591-13
    13659218-13-7
    5918353-724
    18533-724-79
    531224-79103
    3211-79103-182
    2120103-182467
    So our multiplicative inverse is -182 mod 467 ≡ 285
    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
    =  (265 × 198475 × 163 +
       16 × 209216 × 11 +
       428 × 190400 × 285)   mod 88916800
    = 2727241 (mod 88916800)


    So our answer is 2727241 (mod 88916800).


Verification

So we found that x ≡ 2727241
If this is correct, then the following statements (i.e. the original equations) are true:
895x (mod 448) ≡ 183 (mod 448)
907x (mod 425) ≡ 62 (mod 425)
681x (mod 467) ≡ 527 (mod 467)

Let's see whether that's indeed the case if we use x ≡ 2727241.