Question

Examine whether `17/30` is a terminating decimal.

Answer 1

Given: `17/30`

Step-wise calculation:

1. Simplify: gcd(17, 30) = 1, so the fraction is already in simplest form.

2. Factor the denominator: 30 = 2 × 3 × 5. Because the denominator contains a prime factor 3, it is not of the form 2^m × 5ⁿ.

A rational `p/q` (in lowest terms) has a terminating decimal if q = 2^m × 5ⁿ; otherwise it is a non‑terminating repeating decimal.

3. Optional: long division shows the decimal expansion repeats:

17 ÷ 30 = 0.5 (remainder 2), then 20 ÷ 30 = 0.6 (remainder 20) and the remainder 20 repeats, so the digits 6 repeat.

Hence, `17/30 = 0.56666`... = `0.bar56` (0.5 with 6 repeating).

`17/30` is a non‑terminating repeating decimal (decimal form 0.56666...).