A positive integer X is of the form X = 2^a \times 3^b, where a and b are positive integer… — State PSC Exam

A positive integer $X$ is of the form $X = 2^a \times 3^b$ , where $a$ and $b$ are positive integers. It is known that $X$ has exactly $15$ positive factors. If $X$ is a multiple of $9$ but is NOT a multiple of $27$ , which of the following is the exact value of $X$ ?

144Answer
B
324
C
48
D
864

Answer

144

The correct value is determined by setting up the factor formula (a+1)(b+1) = 15. The only valid exponents that satisfy both the positive integer requirement and the divisibility constraints (b < 3 to avoid being a multiple of 27) are a=4 and b=2. Evaluating 2^4 \times 3^2 yields exactly 144.

Step-by-Step Solution

Identify the formula for the total number of positive factors.

For a prime factorization

X = p^a \times q^b

, the number of positive factors is

(a+1)(b+1)

This is a fundamental property of prime factorization.

Set up the equation based on the given number of factors.

(a+1)(b+1) = 15

The problem explicitly states that

X

has exactly 15 positive factors.

Determine the possible integer values for exponents

a

and

b

Since

a \ge 1

and

b \ge 1

, we need factor pairs of 15 that are

\ge 2

. The only pair is

3 \times 5

. This results in two possible cases:

(a+1=5, b+1=3)

which gives

a=4, b=2

; OR

(a+1=3, b+1=5)

which gives

a=2, b=4

Both

a

and

b

must be positive integers as defined in the question stem.

Apply the divisibility constraints to filter the two cases.

The condition 'NOT a multiple of 27' means

3^b

cannot be divisible by

3^3

, meaning

b

must be strictly less than

3

. Therefore,

b

must be

2

, which forces

a

to be

4

b=4

were chosen,

X

would be a multiple of

81

, which is a multiple of

27

, violating the negative constraint.

Calculate the final value of

X

X = 2^4 \times 3^2 = 16 \times 9 = 144

Substitute the verified exponents back into the original prime factorization.

Key Concept

Prime Factorization and Number of Factors

Estimated Time:1m 30s

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