Minimum logic for a MOD-64 synchronous binary counter Which group of logic devices represents the minimum required to implement a MOD-64 synchronous binary counter?

Introduction / Context:
A synchronous binary counter advances all flip-flops in lockstep with the clock, using combinational logic to generate each stage’s toggle condition. Determining the minimum hardware helps optimize resource usage in ASICs, FPGAs, and discrete designs.

Given Data / Assumptions:

• Target modulus is 64 → requires 6 bits since 2^6 = 64.
• We may use multi-input AND gates where helpful (common in practice).
• Counter style is synchronous (not ripple), so toggling is controlled by combinational conditions.

Concept / Approach:
For a binary counter with T-type behavior: T0 = 1 (LSB toggles every clock), T1 = Q0, T2 = Q1 * Q0, T3 = Q2 * Q1 * Q0, T4 = Q3 * Q2 * Q1 * Q0, T5 = Q4 * Q3 * Q2 * Q1 * Q0. Using one multi-input AND per stage that needs a product term yields four AND gates for stages 2 through 5; stages 0 and 1 need none beyond simple wiring.

Step-by-Step Solution:

Verification / Alternative check:
Synthesis tools for FPGAs commonly infer this structure with equivalent LUT usage; discrete logic implementations map directly to available multi-input AND gates.

Why Other Options Are Wrong:

• Five flip-flops: can only reach 2^5 = 32 states (MOD-32).
• Seven flip-flops: unnecessary hardware for MOD-64; also AND count is not minimal.
• Four flip-flops: at most MOD-16.

Common Pitfalls:
Assuming each stage requires many pairwise 2-input ANDs; if multi-input ANDs are available, the count reduces to one per higher stage.

Final Answer:
Six flip-flops, four AND gates