What is the minimum number of flip-flops required to construct a binary…

Concept

A flip-flop stores one binary bit, so k flip-flops provide 2k distinct binary states.

A modulo-N counter must represent N different count states. The smallest usable integer k must therefore satisfy 2k ≥ N.

Application

1. Let k be the minimum number of flip-flops and let N be the required number of counter states.

2. The available state capacity is 2k, so the design condition is 2k ≥ N.

3. Taking the base-2 logarithm gives k ≥ log2 N.

4. Because k must be an integer, its minimum value is k = ⌈log2 N⌉.

Cross-check

1. For N = 5, ⌈log2 N⌉ = 3.

2. Two flip-flops provide 22 = 4 states, which cannot represent all five states.

3. Three flip-flops provide 23 = 8 states, which can represent all five states.

Result

The minimum number of flip-flops is ⌈log2 N⌉.