A MOD 2 and a MOD 5 up-counter when cascaded together results in a MOD _____ counter. (in integer)

The correct answer is 10.

To understand why the cascaded connection of a MOD 2 counter and a MOD 5 counter results in a MOD 10 counter, we can analyze the behavior of cascaded counters step-by-step.

A counter's MOD number (modulus) represents the total number of unique states it can represent before recycling back to its starting state. For example:
- A MOD 2 counter has 2 states (typically representing counts 0 and 1).
- A MOD 5 counter has 5 states (typically representing counts 0, 1, 2, 3, and 4).

When two counters are cascaded, the output of the first stage acts as the clock trigger (or enable signal) for the second stage. This means that for every complete cycle of states in the first stage, the second stage advances by only one state.

Mathematically, the total number of unique states (overall Modulus, $M_{\mathrm{total}}$) for cascaded counters is the product of the individual modulus values of each counter stage:
$M_{\mathrm{total}}=\; M$₁ × M₂

Given:
- Modulus of the first counter, $M_1=\; 2$
- Modulus of the second counter, $M_2=\; 5$

Substituting these values into our formula:
$M_{\mathrm{total}}=\; 2\; \times \; 5\; =\; 10$

Therefore, the cascaded configuration results in a MOD 10 counter, which has 10 unique counting states (typically counting from 0 to 9).