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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?

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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 "on" and "off" states depends on how the outputs of the two dividers are combined (e.g. "and", "or" or "xor" gate).

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 $\text{lcm}(b_1,b_2, ... ,b_n)$.

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