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 |
|---|---|---|---|---|---|---|
| 745 | 474 | 1 | 271 | 0 | 1 | -1 |
| 474 | 271 | 1 | 203 | 1 | -1 | 2 |
| 271 | 203 | 1 | 68 | -1 | 2 | -3 |
| 203 | 68 | 2 | 67 | 2 | -3 | 8 |
| 68 | 67 | 1 | 1 | -3 | 8 | -11 |
| 67 | 1 | 67 | 0 | 8 | -11 | 745 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 941 | 362 | 2 | 217 | 0 | 1 | -2 |
| 362 | 217 | 1 | 145 | 1 | -2 | 3 |
| 217 | 145 | 1 | 72 | -2 | 3 | -5 |
| 145 | 72 | 2 | 1 | 3 | -5 | 13 |
| 72 | 1 | 72 | 0 | -5 | 13 | -941 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 659 | 163 | 4 | 7 | 0 | 1 | -4 |
| 163 | 7 | 23 | 2 | 1 | -4 | 93 |
| 7 | 2 | 3 | 1 | -4 | 93 | -283 |
| 2 | 1 | 2 | 0 | 93 | -283 | 659 |
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 ≡ 406 × 474-1 (mod 745) ≡ 406 × 734 (mod 745) ≡ 4 (mod 745)x ≡ 310 × 362-1 (mod 941) ≡ 310 × 13 (mod 941) ≡ 266 (mod 941)x ≡ 735 × 163-1 (mod 659) ≡ 735 × 376 (mod 659) ≡ 239 (mod 659)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 4 (mod 745)
x ≡ 266 (mod 941)
x ≡ 239 (mod 659)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 745 × 941 × 659 = 461988655 - We calculate the numbers M1 to M3
M1=M/m1=461988655/745=620119,M2=M/m2=461988655/941=490955,M3=M/m3=461988655/659=701045 - 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 -251 mod 745 ≡ 494n b q r t1 t2 t3 745 620119 0 745 0 1 0 620119 745 832 279 1 0 1 745 279 2 187 0 1 -2 279 187 1 92 1 -2 3 187 92 2 3 -2 3 -8 92 3 30 2 3 -8 243 3 2 1 1 -8 243 -251 2 1 2 0 243 -251 745
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 80 mod 941 ≡ 80n b q r t1 t2 t3 941 490955 0 941 0 1 0 490955 941 521 694 1 0 1 941 694 1 247 0 1 -1 694 247 2 200 1 -1 3 247 200 1 47 -1 3 -4 200 47 4 12 3 -4 19 47 12 3 11 -4 19 -61 12 11 1 1 19 -61 80 11 1 11 0 -61 80 -941
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 166 mod 659 ≡ 166n b q r t1 t2 t3 659 701045 0 659 0 1 0 701045 659 1063 528 1 0 1 659 528 1 131 0 1 -1 528 131 4 4 1 -1 5 131 4 32 3 -1 5 -161 4 3 1 1 5 -161 166 3 1 3 0 -161 166 -659
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
= (4 × 620119 × 494 +
266 × 490955 × 80 +
239 × 701045 × 166) mod 461988655
= 217101199 (mod 461988655)
So our answer is 217101199 (mod 461988655).
Verification
So we found that x ≡ 217101199
If this is correct, then the following statements (i.e. the original equations) are true:
474x (mod 745) ≡ 406 (mod 745)
362x (mod 941) ≡ 310 (mod 941)
163x (mod 659) ≡ 735 (mod 659)
Let's see whether that's indeed the case if we use x ≡ 217101199.