Cascading counters — when two modulus-N counters are cascaded, the overall modulus (mod number) of the combined counter equals the ______ of the individual mod numbers.

Introduction / Context:
Complex counting ranges are achieved by cascading counters. Understanding how the combined modulus relates to the individual moduli is fundamental for designing frequency dividers, timers, and sequence generators. This question asks for that relationship.

Given Data / Assumptions:

• Two counters connected in cascade (output of one drives the count-enabling or clock of the next stage appropriately).
• Individual moduli are M and N.
• Each counter cycles through its full count before advancing the next stage appropriately.

Concept / Approach:
In a properly cascaded arrangement, the first counter advances through M states; for each full M-cycle, it increments the second counter by one state, which cycles through N states. The total unique state combinations are M * N, so the overall modulus is the product of the individual moduli.

Step-by-Step Solution:

Verification / Alternative check:
Example: Cascading mod-10 and mod-6 yields mod-60 division, common in clock/timer designs (seconds 0–59).

Why Other Options Are Wrong:

• Sum: Would imply mod (M + N), not consistent with state space combinations.
• Logarithm/Reciprocal: Not meaningful for counting modulus in digital logic.

Common Pitfalls:
Incorrect gating between stages or misinterpreting “ripple carry” signals can break the ideal product relationship; proper cascade wiring restores the M * N behavior.

Final Answer:
Product.