Frequency division using flip-flops in cascade How many flip-flops are required to implement a divide-by-128 counter (i.e., overall division factor of 128)?

Introduction / Context:
Binary ripple and synchronous counters are built by cascading flip-flops, each halving the frequency (divide-by-2) when configured to toggle. The total division achieved is 2^N where N is the number of flip-flops. Determining the required number for a target division is a core competency in digital design and timing generation.

Given Data / Assumptions:

• Each flip-flop provides divide-by-2 when toggled (e.g., J = K = 1 on J–K, T = 1 on T-FF, or clocked D with feedback).
• We need an overall divide-by-128.
• Counter can be ripple or synchronous; the division count relation 2^N holds in either case.

Concept / Approach:

To obtain a division of 2^N = 128, solve for N. Since 2^7 = 128, seven toggle stages are required. Cascading seven flip-flops yields outputs at successively halved frequencies, with the most significant stage output toggling every 128 input cycles.

Step-by-Step Solution:

Verification / Alternative check:

Trace a ripple counter: stage outputs divide by 2, 4, 8, 16, 32, 64, 128. After seven stages, the final output period spans 128 input clocks.

Why Other Options Are Wrong:

1, 4, 6 flip-flops yield 2, 16, 64 division—insufficient.

More than 7 would exceed the needed division (e.g., 8 gives 256).

Common Pitfalls:

Confusing the number of states (2^N) with the maximum count value; forgetting that unused higher counts must be decoded and reset in non-power-of-two modulus counters, which is not needed here.

Final Answer:

7