Worked by hand
Examples you can check tick by tick
Every example uses one abstract integer-tick unit and origin. Copy the two or three rows into the calculator; there is no header.
Two offset patterns: 17 modulo 35
5,2 7,3
The first cycle includes 2, 7, 12, 17, 22 … . The second includes 3, 10, 17, 24 … . Their first nonnegative common tick is 17. Periods 5 and 7 are coprime, so the exact repeat is 35. The first four alignments are 17, 52, 87 and 122. Check: 17 − 2 = 15 is divisible by 5, and 17 − 3 = 14 is divisible by 7.
Shared factors can still work
6,2 8,4
After the first row, the combined residue is 2 modulo 6. The difference for the next phase is 4 − 2 = 2, divisible by gcd(6,8) = 2. A common tick exists. The first is 20, and the repeat is 6 × (8/2) = 24. The first four are 20, 44, 68 and 92. Both 20 − 2 = 18 and 20 − 4 = 16 divide exactly by their respective periods.
Valid rows, but no common tick
4,0 6,1
Row 2 conflicts with the combined earlier constraints. The difference is 1 − 0 = 1, which is not divisible by gcd(4,6) = 2. The first row permits only even ticks; the second permits only odd ticks. The result is no common integer tick. You can still save this current result and its exact reason.
A repeated factor can narrow the residue
4,1 8,5
Period 8 is a multiple of 4, and phase 5 modulo 8 is also phase 1 modulo 4. The common ticks are 5, 13, 21 and 29, with repeat 8. Repeating the same input row would add no restriction and would not change this answer.
A negative phase wraps to its residue
3,-1 4,0
Phase −1 modulo 3 normalizes to 2. The least nonnegative meeting with phase 0 modulo 4 is 8, then 20, 32 and 44, repeating every 12 ticks. The normalization changes the displayed representative, not the set of allowed ticks.
The origin is a real member of the model
3,0 4,0 5,0
All three congruences include tick 0. The least nonnegative alignment is therefore 0; the repeat is 60. The first four are 0, 60, 120 and 180. If you want the next strictly positive alignment for this example, it is 60. This distinction does not turn the abstract ticks into clock times.
Period 1 adds no restriction
1,99999999 7,3
Every integer has residue 0 modulo 1, so the first entered phase normalizes to 0. The second row determines the result: 3, 10, 17 and 24, with repeat 7. The large entered phase is not treated as a first-ever starting tick.
The tool checks exact ideal patterns, not measured timing. Real cycle drift, clock synchronization, timezones, activation dates and safety requirements need a different model.