ITPEC FE Morning April 2021 Question 2

Finite Octal Fractions — determine which decimal becomes a finite octal fraction.

To check if a decimal has a finite representation in base 8, multiply the fractional part by 8 repeatedly:

• •0.5: 0.5 × 8 = 4.0 → terminates in 1 step → 0.4₈ ✅
• •0.3: 0.3 × 8 = 2.4 → 0.4 × 8 = 3.2 → ... → repeats ❌
• •0.4: 0.4 × 8 = 3.2 → 0.2 × 8 = 1.6 → ... → repeats ❌
• •0.8: 0.8 × 8 = 6.4 → 0.4 × 8 = 3.2 → ... → repeats ❌

Key rule: A fraction p/q (in lowest terms) terminates in base b only if all prime factors of q are also prime factors of b. Since 8 = 2³, the denominator must be a power of 2.

• •0.5 = 1/2 → denominator is 2 → ✅
• •0.3 = 3/10, 0.4 = 2/5, 0.8 = 4/5 → denominator contains 5 → ❌

AI-generated — may contain errors