ITPEC FE Morning April 2021 Question 2

0.5 — it is 1 ÷ 2, and a fraction ends in base 8 only when its denominator has no prime factor other than 2.

Multiply the fraction by 8 and keep the fractional part until it reaches 0:

• •0.5: 0.5 × 8 = 4.0, so 0.5 = 0.4 in octal, finished in one step
• •0.3: 0.3 × 8 = 2.4, 0.4 × 8 = 3.2, 0.2 × 8 = 1.6, 0.6 × 8 = 4.8, 0.8 × 8 = 6.4, and 0.4 comes back, so the digits repeat

In lowest terms 0.3 = 3 ÷ 10, 0.4 = 2 ÷ 5 and 0.8 = 4 ÷ 5: each denominator contains the factor 5, which 8 does not have, so none of them can end.

Answer (c)

Why not others:

• •(a) 0.3 — 3 ÷ 10; the factor 5 in 10 makes its octal digits 0.23146314… go on forever
• •(b) 0.4 — 2 ÷ 5; its octal digits 0.3146 repeat
• •(d) 0.8 — 4 ÷ 5; its octal digits 0.6314 repeat

Key rule: A fraction in lowest terms ends in base b only if every prime factor of its denominator divides b; for bases 2, 8 and 16 the denominator must be a power of 2.

Basic Theory · AI-written, may contain errors