If both and are factors of the number , then what is the smallest possible value of ?

Explanation: 1. Prime Factorization of the Expression: 2. Identifying Required Factors: For : The expression already has . Since , this condition is satisfied. For : There is no in the expression. Therefore, must provide . 3. Matching with Given Options: Since is not in the options, we check the smallest multiple of among the choices: (a) (b) (c) (Not divisible by 49) (d) The smallest value among the options that contains is . Final Answer: Correct Option: (b) 147

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NIMCET 2008 — Reasoning PYQ

NIMCET | Reasoning | 2008

If both $7^2$ and $3^3$ are factors of the number $(a \cdot 11^3 \cdot 6^2 \cdot 3^{11})$ , then what is the smallest possible value of $a$ ?

Choose the correct answer:

Explanation

1. Prime Factorization of the Expression:

N = a \cdot 11^3 \cdot 6^2 \cdot 3^{11}

N = a \cdot 11^3 \cdot (2 \cdot 3)^2 \cdot 3^{11}

N = a \cdot 11^3 \cdot 2^2 \cdot 3^2 \cdot 3^{11}

N = a \cdot 2^2 \cdot 3^{13} \cdot 11^3

2. Identifying Required Factors:

For $3^3$ : The expression already has $3^{13}$ . Since $13 \ge 3$ , this condition is satisfied.
For $7^2$ : There is no $7$ in the expression. Therefore, $a$ must provide $7^2$ .

$a_{min} = 7^2 = 49$

3. Matching with Given Options:

Since $49$ is not in the options, we check the smallest multiple of $49$ among the choices:

(a) $1323 = 49 \times 27$
(b) $147 = 49 \times 3$
(c) $21$ (Not divisible by 49)
(d) $3087 = 49 \times 63$

The smallest value among the options that contains $7^2$ is $147$ .

Final Answer:

Correct Option: (b) 147