# Deriving simple formula for summation

I am first year non-math student and I am trying to formalize my homework from digital circuits course using some basic math tools.

To give some initial context:

If I have $$\frac{1}{5}$$ frequency divider it means: $$1$$ 'on' state and $$4$$ 'off' states. (total number of states $$= 5$$)

If I have $$\frac{2}{11}$$ frequency divider it means: $$2$$ 'on' states and $$9$$ 'off' states. (total number of states $$= 11$$)

So if I have only 1 frequency divider $$\frac{a}{b}$$ total number of states:

$$\frac{a}{b} \quad \land \quad a, b \in \mathbb{N} \quad \land \quad a,b \neq 0\quad \Longrightarrow \quad T_{S} = b$$

where: $$T_{S}$$ denotes total number of states.

That is fairly simple.

Now, if I have two frequency dividers, for example:

$$\frac{12}{13}$$ and $$\frac{9}{50}$$ then the total number of states equals the maximal number of 'on' states from all dividers (here: two dividers) + the maximal number of 'off' states from all dividers (here: two dividers).

$$T_{S} = \max{(S_{on}^{1} + S_{on}^{2})} + \max{(S_{off}^{1} + S_{off}^{2})}$$

$$T_{S} = \max{(12, 9)} + \max{(1, 41)}$$

$$T_{S} = 12 + 41$$

$$T_{S} = 53$$

My questions are:

Can someone help me further formalize this formula? I just want it not to have any mistakes regarding both logic and "math syntax".

I would be grateful if someone could derive formula for $$n$$ number of dividers, probably using some $$\sum$$ symbol. I know how to calculate it for $$n$$ number of dividers, I just dont know how to make a formula out of it so it does make sense in a math way.

Really not sure how to tag this thread.

Actually I came up with this, not sure it makes any sense?

Let $$n$$ denote positive natual number of frequency dividers:

$$\frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, ..., \frac{a_{n-1}}{b_{n-1}}, \frac{a_{n}}{b_{n}}$$

The total number of states $$T_{S}$$ is:

$$T_{S} = \max{(a_{1}, a_{2}, ..., a_{n - 1}, a_{n})} + \max{(b_{1}, b_{2}, ..., b_{n - 1}, b_{n})}$$

Does it make any sense? Does it have mistakes?

Assuming both dividers are synchronized running off the same clock, the total number of states assuming a $$\frac{12}{13}$$ and $$\frac{9}{50}$$ divider would be $$\text{lcm}(13,50)=650$$.
The total number of states assuming $$\frac{a_1}{b_1}$$, $$\frac{a_2}{b_2}$$, ... , and $$\frac{a_n}{b_n}$$ dividers would be $$\text{lcm}(b_1,b_2, ... ,b_n)$$.