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.
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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 |
|---|---|---|---|---|---|---|
| 781 | 995 | 0 | 781 | 0 | 1 | 0 |
| 995 | 781 | 1 | 214 | 1 | 0 | 1 |
| 781 | 214 | 3 | 139 | 0 | 1 | -3 |
| 214 | 139 | 1 | 75 | 1 | -3 | 4 |
| 139 | 75 | 1 | 64 | -3 | 4 | -7 |
| 75 | 64 | 1 | 11 | 4 | -7 | 11 |
| 64 | 11 | 5 | 9 | -7 | 11 | -62 |
| 11 | 9 | 1 | 2 | 11 | -62 | 73 |
| 9 | 2 | 4 | 1 | -62 | 73 | -354 |
| 2 | 1 | 2 | 0 | 73 | -354 | 781 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 151 | 878 | 0 | 151 | 0 | 1 | 0 |
| 878 | 151 | 5 | 123 | 1 | 0 | 1 |
| 151 | 123 | 1 | 28 | 0 | 1 | -1 |
| 123 | 28 | 4 | 11 | 1 | -1 | 5 |
| 28 | 11 | 2 | 6 | -1 | 5 | -11 |
| 11 | 6 | 1 | 5 | 5 | -11 | 16 |
| 6 | 5 | 1 | 1 | -11 | 16 | -27 |
| 5 | 1 | 5 | 0 | 16 | -27 | 151 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 779 | 63 | 12 | 23 | 0 | 1 | -12 |
| 63 | 23 | 2 | 17 | 1 | -12 | 25 |
| 23 | 17 | 1 | 6 | -12 | 25 | -37 |
| 17 | 6 | 2 | 5 | 25 | -37 | 99 |
| 6 | 5 | 1 | 1 | -37 | 99 | -136 |
| 5 | 1 | 5 | 0 | 99 | -136 | 779 |
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 ≡ 840 × 995-1 (mod 781) ≡ 840 × 427 (mod 781) ≡ 201 (mod 781)x ≡ 205 × 878-1 (mod 151) ≡ 205 × 124 (mod 151) ≡ 52 (mod 151)x ≡ 42 × 63-1 (mod 779) ≡ 42 × 643 (mod 779) ≡ 520 (mod 779)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 201 (mod 781)
x ≡ 52 (mod 151)
x ≡ 520 (mod 779)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 781 × 151 × 779 = 91868249 - We calculate the numbers M1 to M3
M1=M/m1=91868249/781=117629,M2=M/m2=91868249/151=608399,M3=M/m3=91868249/779=117931 - 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 -75 mod 781 ≡ 706n b q r t1 t2 t3 781 117629 0 781 0 1 0 117629 781 150 479 1 0 1 781 479 1 302 0 1 -1 479 302 1 177 1 -1 2 302 177 1 125 -1 2 -3 177 125 1 52 2 -3 5 125 52 2 21 -3 5 -13 52 21 2 10 5 -13 31 21 10 2 1 -13 31 -75 10 1 10 0 31 -75 781
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 68 mod 151 ≡ 68n b q r t1 t2 t3 151 608399 0 151 0 1 0 608399 151 4029 20 1 0 1 151 20 7 11 0 1 -7 20 11 1 9 1 -7 8 11 9 1 2 -7 8 -15 9 2 4 1 8 -15 68 2 1 2 0 -15 68 -151
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -276 mod 779 ≡ 503n b q r t1 t2 t3 779 117931 0 779 0 1 0 117931 779 151 302 1 0 1 779 302 2 175 0 1 -2 302 175 1 127 1 -2 3 175 127 1 48 -2 3 -5 127 48 2 31 3 -5 13 48 31 1 17 -5 13 -18 31 17 1 14 13 -18 31 17 14 1 3 -18 31 -49 14 3 4 2 31 -49 227 3 2 1 1 -49 227 -276 2 1 2 0 227 -276 779
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
= (201 × 117629 × 706 +
52 × 608399 × 68 +
520 × 117931 × 503) mod 91868249
= 80737638 (mod 91868249)
So our answer is 80737638 (mod 91868249).
Verification
So we found that x ≡ 80737638
If this is correct, then the following statements (i.e. the original equations) are true:
995x (mod 781) ≡ 840 (mod 781)
878x (mod 151) ≡ 205 (mod 151)
63x (mod 779) ≡ 42 (mod 779)
Let's see whether that's indeed the case if we use x ≡ 80737638.