sequences and series - arithmetic progression, division and remainder. finding pattern length

When all elements in a arithmetic progression are divided by a constant number k, and are written down the remainder, you'll quickly notice that a series of numbers will appear. simple example

7   42  77  112 147 182 217 252 287 322 357 392 (progression, 35 added each time)

7   42  17  52  27  2   37  12  47  22  57  32  (remainder after division by 60) 

When 35 is added again, that gives 392+35=427. The remainder after division by 60 is again 7. The pattern

7   42  17  52  27  2   37  12  47  22  57  32

repeats itself. In this example, this pattern consists of 12 numbers.

Can you calculate the length of that pattern with a formula when you know the constant difference in the progression and the constant number? I tried to figure this out, but failed.

Answer

The period in the example is the smallest number $k$ with the property $$35k\equiv 0 \mod 60$$

We get $$k=\frac{60}{\gcd(35,60)}=\frac{60}{5}=12$$

This way you can find the period in general.