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 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 |
|---|---|---|---|---|---|---|
| 659 | 847 | 0 | 659 | 0 | 1 | 0 |
| 847 | 659 | 1 | 188 | 1 | 0 | 1 |
| 659 | 188 | 3 | 95 | 0 | 1 | -3 |
| 188 | 95 | 1 | 93 | 1 | -3 | 4 |
| 95 | 93 | 1 | 2 | -3 | 4 | -7 |
| 93 | 2 | 46 | 1 | 4 | -7 | 326 |
| 2 | 1 | 2 | 0 | -7 | 326 | -659 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 359 | 376 | 0 | 359 | 0 | 1 | 0 |
| 376 | 359 | 1 | 17 | 1 | 0 | 1 |
| 359 | 17 | 21 | 2 | 0 | 1 | -21 |
| 17 | 2 | 8 | 1 | 1 | -21 | 169 |
| 2 | 1 | 2 | 0 | -21 | 169 | -359 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 428 | 15 | 28 | 8 | 0 | 1 | -28 |
| 15 | 8 | 1 | 7 | 1 | -28 | 29 |
| 8 | 7 | 1 | 1 | -28 | 29 | -57 |
| 7 | 1 | 7 | 0 | 29 | -57 | 428 |
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 ≡ 444 × 847-1 (mod 659) ≡ 444 × 326 (mod 659) ≡ 423 (mod 659)x ≡ 956 × 376-1 (mod 359) ≡ 956 × 169 (mod 359) ≡ 14 (mod 359)x ≡ 771 × 15-1 (mod 428) ≡ 771 × 371 (mod 428) ≡ 137 (mod 428)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 423 (mod 659)
x ≡ 14 (mod 359)
x ≡ 137 (mod 428)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 659 × 359 × 428 = 101256668 - We calculate the numbers M1 to M3
M1=M/m1=101256668/659=153652,M2=M/m2=101256668/359=282052,M3=M/m3=101256668/428=236581 - 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 -182 mod 659 ≡ 477n b q r t1 t2 t3 659 153652 0 659 0 1 0 153652 659 233 105 1 0 1 659 105 6 29 0 1 -6 105 29 3 18 1 -6 19 29 18 1 11 -6 19 -25 18 11 1 7 19 -25 44 11 7 1 4 -25 44 -69 7 4 1 3 44 -69 113 4 3 1 1 -69 113 -182 3 1 3 0 113 -182 659
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -103 mod 359 ≡ 256n b q r t1 t2 t3 359 282052 0 359 0 1 0 282052 359 785 237 1 0 1 359 237 1 122 0 1 -1 237 122 1 115 1 -1 2 122 115 1 7 -1 2 -3 115 7 16 3 2 -3 50 7 3 2 1 -3 50 -103 3 1 3 0 50 -103 359
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is -187 mod 428 ≡ 241n b q r t1 t2 t3 428 236581 0 428 0 1 0 236581 428 552 325 1 0 1 428 325 1 103 0 1 -1 325 103 3 16 1 -1 4 103 16 6 7 -1 4 -25 16 7 2 2 4 -25 54 7 2 3 1 -25 54 -187 2 1 2 0 54 -187 428
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
= (423 × 153652 × 477 +
14 × 282052 × 256 +
137 × 236581 × 241) mod 101256668
= 30716413 (mod 101256668)
So our answer is 30716413 (mod 101256668).
Verification
So we found that x ≡ 30716413
If this is correct, then the following statements (i.e. the original equations) are true:
847x (mod 659) ≡ 444 (mod 659)
376x (mod 359) ≡ 956 (mod 359)
15x (mod 428) ≡ 771 (mod 428)
Let's see whether that's indeed the case if we use x ≡ 30716413.