Welcome to ChineseRemainderTheorem.com!
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.
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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?- Read about how to execute the Chinese Remainder Theorem
- Discover different ways to calculate multiplicative inverses
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:
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 521 | 804 | 0 | 521 | 0 | 1 | 0 |
| 804 | 521 | 1 | 283 | 1 | 0 | 1 |
| 521 | 283 | 1 | 238 | 0 | 1 | -1 |
| 283 | 238 | 1 | 45 | 1 | -1 | 2 |
| 238 | 45 | 5 | 13 | -1 | 2 | -11 |
| 45 | 13 | 3 | 6 | 2 | -11 | 35 |
| 13 | 6 | 2 | 1 | -11 | 35 | -81 |
| 6 | 1 | 6 | 0 | 35 | -81 | 521 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 888 | 425 | 2 | 38 | 0 | 1 | -2 |
| 425 | 38 | 11 | 7 | 1 | -2 | 23 |
| 38 | 7 | 5 | 3 | -2 | 23 | -117 |
| 7 | 3 | 2 | 1 | 23 | -117 | 257 |
| 3 | 1 | 3 | 0 | -117 | 257 | -888 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 151 | 737 | 0 | 151 | 0 | 1 | 0 |
| 737 | 151 | 4 | 133 | 1 | 0 | 1 |
| 151 | 133 | 1 | 18 | 0 | 1 | -1 |
| 133 | 18 | 7 | 7 | 1 | -1 | 8 |
| 18 | 7 | 2 | 4 | -1 | 8 | -17 |
| 7 | 4 | 1 | 3 | 8 | -17 | 25 |
| 4 | 3 | 1 | 1 | -17 | 25 | -42 |
| 3 | 1 | 3 | 0 | 25 | -42 | 151 |
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 ≡ 542 × 804-1 (mod 521) ≡ 542 × 440 (mod 521) ≡ 383 (mod 521)x ≡ 635 × 425-1 (mod 888) ≡ 635 × 257 (mod 888) ≡ 691 (mod 888)x ≡ 49 × 737-1 (mod 151) ≡ 49 × 109 (mod 151) ≡ 56 (mod 151)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 383 (mod 521)
x ≡ 691 (mod 888)
x ≡ 56 (mod 151)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 521 × 888 × 151 = 69859848 - We calculate the numbers M1 to M3
M1=M/m1=69859848/521=134088,M2=M/m2=69859848/888=78671,M3=M/m3=69859848/151=462648 - 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:
So our multiplicative inverse is -30 mod 521 ≡ 491n b q r t1 t2 t3 521 134088 0 521 0 1 0 134088 521 257 191 1 0 1 521 191 2 139 0 1 -2 191 139 1 52 1 -2 3 139 52 2 35 -2 3 -8 52 35 1 17 3 -8 11 35 17 2 1 -8 11 -30 17 1 17 0 11 -30 521
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -337 mod 888 ≡ 551n b q r t1 t2 t3 888 78671 0 888 0 1 0 78671 888 88 527 1 0 1 888 527 1 361 0 1 -1 527 361 1 166 1 -1 2 361 166 2 29 -1 2 -5 166 29 5 21 2 -5 27 29 21 1 8 -5 27 -32 21 8 2 5 27 -32 91 8 5 1 3 -32 91 -123 5 3 1 2 91 -123 214 3 2 1 1 -123 214 -337 2 1 2 0 214 -337 888
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 66 mod 151 ≡ 66n b q r t1 t2 t3 151 462648 0 151 0 1 0 462648 151 3063 135 1 0 1 151 135 1 16 0 1 -1 135 16 8 7 1 -1 9 16 7 2 2 -1 9 -19 7 2 3 1 9 -19 66 2 1 2 0 -19 66 -151
Source: ExtendedEuclideanAlgorithm.com
- 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
= (383 × 134088 × 491 +
691 × 78671 × 551 +
56 × 462648 × 66) mod 69859848
= 12956611 (mod 69859848)
So our answer is 12956611 (mod 69859848).
Verification
So we found that x ≡ 12956611
If this is correct, then the following statements (i.e. the original equations) are true:
804x (mod 521) ≡ 542 (mod 521)
425x (mod 888) ≡ 635 (mod 888)
737x (mod 151) ≡ 49 (mod 151)
Let's see whether that's indeed the case if we use x ≡ 12956611.