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" .
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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 |
|---|---|---|---|---|---|---|
| 701 | 560 | 1 | 141 | 0 | 1 | -1 |
| 560 | 141 | 3 | 137 | 1 | -1 | 4 |
| 141 | 137 | 1 | 4 | -1 | 4 | -5 |
| 137 | 4 | 34 | 1 | 4 | -5 | 174 |
| 4 | 1 | 4 | 0 | -5 | 174 | -701 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 541 | 87 | 6 | 19 | 0 | 1 | -6 |
| 87 | 19 | 4 | 11 | 1 | -6 | 25 |
| 19 | 11 | 1 | 8 | -6 | 25 | -31 |
| 11 | 8 | 1 | 3 | 25 | -31 | 56 |
| 8 | 3 | 2 | 2 | -31 | 56 | -143 |
| 3 | 2 | 1 | 1 | 56 | -143 | 199 |
| 2 | 1 | 2 | 0 | -143 | 199 | -541 |
Source: ExtendedEuclideanAlgorithm.com
| n | b | q | r | t1 | t2 | t3 |
|---|---|---|---|---|---|---|
| 979 | 899 | 1 | 80 | 0 | 1 | -1 |
| 899 | 80 | 11 | 19 | 1 | -1 | 12 |
| 80 | 19 | 4 | 4 | -1 | 12 | -49 |
| 19 | 4 | 4 | 3 | 12 | -49 | 208 |
| 4 | 3 | 1 | 1 | -49 | 208 | -257 |
| 3 | 1 | 3 | 0 | 208 | -257 | 979 |
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 ≡ 55 × 560-1 (mod 701) ≡ 55 × 174 (mod 701) ≡ 457 (mod 701)x ≡ 545 × 87-1 (mod 541) ≡ 545 × 199 (mod 541) ≡ 255 (mod 541)x ≡ 176 × 899-1 (mod 979) ≡ 176 × 722 (mod 979) ≡ 781 (mod 979)
So now we have the following equations of the form x ≡ a (mod m):
x ≡ 457 (mod 701)
x ≡ 255 (mod 541)
x ≡ 781 (mod 979)
Now the actual calculation
- Find the common modulus M
M = m1 × m2 × ... × mk = 701 × 541 × 979 = 371276939 - We calculate the numbers M1 to M3
M1=M/m1=371276939/701=529639,M2=M/m2=371276939/541=686279,M3=M/m3=371276939/979=379241 - 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 -272 mod 701 ≡ 429n b q r t1 t2 t3 701 529639 0 701 0 1 0 529639 701 755 384 1 0 1 701 384 1 317 0 1 -1 384 317 1 67 1 -1 2 317 67 4 49 -1 2 -9 67 49 1 18 2 -9 11 49 18 2 13 -9 11 -31 18 13 1 5 11 -31 42 13 5 2 3 -31 42 -115 5 3 1 2 42 -115 157 3 2 1 1 -115 157 -272 2 1 2 0 157 -272 701
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 132 mod 541 ≡ 132n b q r t1 t2 t3 541 686279 0 541 0 1 0 686279 541 1268 291 1 0 1 541 291 1 250 0 1 -1 291 250 1 41 1 -1 2 250 41 6 4 -1 2 -13 41 4 10 1 2 -13 132 4 1 4 0 -13 132 -541
Source: ExtendedEuclideanAlgorithm.com
So our multiplicative inverse is 141 mod 979 ≡ 141n b q r t1 t2 t3 979 379241 0 979 0 1 0 379241 979 387 368 1 0 1 979 368 2 243 0 1 -2 368 243 1 125 1 -2 3 243 125 1 118 -2 3 -5 125 118 1 7 3 -5 8 118 7 16 6 -5 8 -133 7 6 1 1 8 -133 141 6 1 6 0 -133 141 -979
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
= (457 × 529639 × 429 +
255 × 686279 × 132 +
781 × 379241 × 141) mod 371276939
= 140133862 (mod 371276939)
So our answer is 140133862 (mod 371276939).
Verification
So we found that x ≡ 140133862
If this is correct, then the following statements (i.e. the original equations) are true:
560x (mod 701) ≡ 55 (mod 701)
87x (mod 541) ≡ 545 (mod 541)
899x (mod 979) ≡ 176 (mod 979)
Let's see whether that's indeed the case if we use x ≡ 140133862.