NIMCET 2007 — Computer PYQ

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Question

NIMCET 2007 — Computer PYQ

NIMCET | Computer | 2007

The number 1/3 can be represented with finite number of terms in base :

Choose the correct answer:

Explanation

The Fundamental Rule of Terminating Fractions

A simplified rational fraction $\frac{p}{q}$ (where $g cd (p, q) = 1$ ) can be expressed as a terminating (finite) fractional expansion in a given base $b$ if and only if all the prime factors of the denominator $q$ are also prime factors of the base $b$ .

Mathematically:

$If q = p_{1}^{e_{1}} \cdot p_{2}^{e_{2}} \dots p_{k}^{e_{k}}$

Then for a fraction to terminate in base $b$ , every prime factor $p_{i}$ must perfectly divide $b$ :

$p_{i} ∣ b \forall i \in {1, 2, \dots, k}$

If the base $b$ does not share all the prime factors of $q$ , the fractional representation will result in an infinite recurring (repeating) sequence.

Step 1: Analyze the Given Fraction

We are given the fraction:

$\frac{p}{q} = \frac{1}{3}$

Here, the denominator is $q = 3$ .
The only prime factor of the denominator is 3.

Therefore, for $\frac{1}{3}$ to have a finite (terminating) representation, the base $b$ must be a multiple of 3.

$3 ∣ b ⟹ b \in {3, 6, 9, 12, 15, \dots}$

Step 2: Test the Given Options

(a) Base 7: The prime factors of $7$ are ${7}$ . Since $3$ does not divide $7$ , the expansion is infinite recurring.
$(\frac{1}{3})_{7} = 0. \overline{2222 \dots}_{7}$
(b) Base 8: The prime factor of $8$ is ${2}$ ( $8 = 2^{3}$ ). Since $3$ does not divide $8$ , the expansion is infinite recurring.
$(\frac{1}{3})_{8} = 0. \overline{2525 \dots}_{8}$
(c) Base 9: The prime factor of $9$ is ${3}$ ( $9 = 3^{2}$ ). Since the prime factor $3$ divides $9$ perfectly ( $3 ∣ 9$ ), the expansion will be finite (terminating).
Let's calculate it:
$\frac{1}{3} = \frac{3}{9} = 3 \times 9^{- 1} = (0.3)_{9}$
Since it terminates in exactly one digit, it contains a finite number of terms.
(d) Base 10: The prime factors of $10$ are ${2, 5}$ . Since $3$ does not divide $10$ , the expansion is infinite recurring.
$(\frac{1}{3})_{10} = 0.3333 \dots_{10} = 0. \overline{3}$

Conclusion

The number $1/3$ has a finite representation only in base 9 among the given choices.

Correct Answer: (c) 9