A rational number $\frac{p}{q}$ in lowest terms has the decimal expansion 2.4576, which stops there. What can you conclude about $q$ ?

Class 10Real NumbersNumber Systems

Aq must be a prime number
Bq has at least one prime factor other than 2 and 5
Cq is of the form $2^{m} \times 5^{n}$ for some non-negative integers $m$ and $n$
Dq must be an odd number

Step-by-Step Solution

The theorem works both ways: a lowest-terms rational terminates IF AND ONLY IF its denominator is 2ᵐ × 5ⁿ. Here

2.4576 = \frac{24576}{10000} = \frac{1536}{625}

, so

q = 625 = 5^{4}

— of the required form, with

m = 0

. A denominator carrying any other prime would force a recurring expansion instead.

Answer: C) q is of the form

2^{m} \times 5^{n}

for some non-negative integers

m

and

n

**Theorem**: A rational number

\frac{p}{q}

(in lowest terms) has a terminating decimal expansion if and only if

q

has the form **2ᵐ × 5ⁿ** (m, n ≥ 0).

**Examples**:
-

\frac{3}{8} : q = 8 = 2^{3}

→ terminates

. (\frac{3}{8} = 0.375.) - \frac{7}{40} : q = 40 = 2^{3} \times 5

→ terminates

. (\frac{7}{40} = 0.175.) - \frac{1}{3} : q = 3

→ does NOT terminate

. (\frac{1}{3} = 0.333

… recurring.)
-

\frac{4}{15} : q = 15 = 3 \times 5

→ does NOT terminate (because of the 3).

**Why**: any decimal expansion is a fraction with denominator 10ⁿ = 2ⁿ × 5ⁿ. To convert

\frac{p}{q}

to decimal form, you need

q

to ONLY have 2s and 5s as primes — otherwise you can't multiply numerator/denominator into a power of 10.

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