Counter terminology — modulus vs maximum count value: For a digital counter, the modulus (mod-N) equals the number of unique states it cycles through (N). Is it correct to say the modulus is the same as the counter’s maximum count capacity N (with states 0 to N-1)?

Introduction / Context:Students frequently mix up two related ideas: the modulus N (the number of distinct states) and the highest numeric code reached (which is N-1 for a simple up-counter starting at 0). Clarifying this relationship helps when designing divide-by-N frequency counters and decoding logic for terminal counts.

Given Data / Assumptions:

• A mod-N counter cycles through N distinct states.
• For a natural binary up-counter, states typically enumerate 0, 1, …, N-1.
• On reaching N-1, the next clock wraps back to 0, completing the cycle.

Concept / Approach:By definition, the modulus equals the period in states before the sequence repeats. Therefore a mod-10 counter repeats every 10 states (0 through 9). While the maximum numeric code observed is N-1, the counter’s “maximum count capacity” in terms of how many unique states it will visit per cycle is indeed N, which is its modulus. This definition is technology-independent and applies to synchronous and asynchronous counters alike.

Step-by-Step Solution:

Verification / Alternative check:Frequency division verifies the same: a mod-N counter divides the input clock by N (one output cycle per N input clocks), matching the definition that N is the number of states per cycle.

Why Other Options Are Wrong:

• Incorrect: Confuses maximum numeric code (N-1) with the count of states (N).
• True only for synchronous counters: Applies equally to ripple and synchronous counters.
• Valid only when counting down: Direction does not change modulus.
• Depends on unused states being masked: Unused states are a design detail; modulus still counts the recurring legal states.

Common Pitfalls:Equating the largest displayed number with modulus; forgetting that 0 is a valid state; assuming modulus depends on implementation type rather than the state machine cycle length.

Final Answer:Correct