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 |
|---|---|---|---|---|---|---|
| 349 | 694 | 0 | 349 | 0 | 1 | 0 |
| 694 | 349 | 1 | 345 | 1 | 0 | 1 |
| 349 | 345 | 1 | 4 | 0 | 1 | -1 |
| 345 | 4 | 86 | 1 | 1 | -1 | 87 |
| 4 | 1 | 4 | 0 | -1 | 87 | -349 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 376 | 89 | 4 | 20 | 0 | 1 | -4 |
| 89 | 20 | 4 | 9 | 1 | -4 | 17 |
| 20 | 9 | 2 | 2 | -4 | 17 | -38 |
| 9 | 2 | 4 | 1 | 17 | -38 | 169 |
| 2 | 1 | 2 | 0 | -38 | 169 | -376 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 667 | 259 | 2 | 149 | 0 | 1 | -2 |
| 259 | 149 | 1 | 110 | 1 | -2 | 3 |
| 149 | 110 | 1 | 39 | -2 | 3 | -5 |
| 110 | 39 | 2 | 32 | 3 | -5 | 13 |
| 39 | 32 | 1 | 7 | -5 | 13 | -18 |
| 32 | 7 | 4 | 4 | 13 | -18 | 85 |
| 7 | 4 | 1 | 3 | -18 | 85 | -103 |
| 4 | 3 | 1 | 1 | 85 | -103 | 188 |
| 3 | 1 | 3 | 0 | -103 | 188 | -667 |
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 ≡ 914 × 694-1 (mod 349) ≡ 914 × 87 (mod 349) ≡ 295 (mod 349)x ≡ 279 × 89-1 (mod 376) ≡ 279 × 169 (mod 376) ≡ 151 (mod 376)x ≡ 157 × 259-1 (mod 667) ≡ 157 × 188 (mod 667) ≡ 168 (mod 667)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 295 (mod 349)
x ≡ 151 (mod 376)
x ≡ 168 (mod 667)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 349 × 376 × 667 = 87526408 - We calculate the numbers M1 to M3
M1=M/m1=87526408/349=250792,M2=M/m2=87526408/376=232783,M3=M/m3=87526408/667=131224 - 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 118 mod 349 ≡ 118n b q r t1 t2 t3 349 250792 0 349 0 1 0 250792 349 718 210 1 0 1 349 210 1 139 0 1 -1 210 139 1 71 1 -1 2 139 71 1 68 -1 2 -3 71 68 1 3 2 -3 5 68 3 22 2 -3 5 -113 3 2 1 1 5 -113 118 2 1 2 0 -113 118 -349
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 135 mod 376 ≡ 135n b q r t1 t2 t3 376 232783 0 376 0 1 0 232783 376 619 39 1 0 1 376 39 9 25 0 1 -9 39 25 1 14 1 -9 10 25 14 1 11 -9 10 -19 14 11 1 3 10 -19 29 11 3 3 2 -19 29 -106 3 2 1 1 29 -106 135 2 1 2 0 -106 135 -376
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 202 mod 667 ≡ 202n b q r t1 t2 t3 667 131224 0 667 0 1 0 131224 667 196 492 1 0 1 667 492 1 175 0 1 -1 492 175 2 142 1 -1 3 175 142 1 33 -1 3 -4 142 33 4 10 3 -4 19 33 10 3 3 -4 19 -61 10 3 3 1 19 -61 202 3 1 3 0 -61 202 -667
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
= (295 × 250792 × 118 +
151 × 232783 × 135 +
168 × 131224 × 202) mod 87526408
= 73181407 (mod 87526408)
So our answer is 73181407 (mod 87526408).
Verification
So we found that x ≡ 73181407
If this is correct, then the following statements (i.e. the original equations) are true:
694x (mod 349) ≡ 914 (mod 349)
89x (mod 376) ≡ 279 (mod 376)
259x (mod 667) ≡ 157 (mod 667)
Let's see whether that's indeed the case if we use x ≡ 73181407.