If p/q is in lowest form and q = 2^m × 5^n × 7^r, where r > 0, what type of decimal expansion will it have?

Non-terminating recurring

The governing rule is that a rational number p/q in lowest form has a terminating decimal expansion only when the prime factors of q are 2 and/or 5. Here q contains 7^r, and r > 0, so at least one factor 7 remains in the reduced denominator. No cancellation with p is possible because p/q is already in lowest form. A denominator containing another prime factor cannot be converted into a power of 10, so the division continues indefinitely. Since the number is rational, its repeating remainder pattern must eventually recur. Therefore option B, non-terminating recurring, is correct. Option A would apply only if no factor other than 2 or 5 remained; option C describes an irrational decimal; option D incorrectly makes termination depend on m and n being equal.

This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Decimal expansion of rational numbers.