Directions: Below question is followed by two statements labelled I and II. Decide if these statements are sufficient to conclusively answer the question. Choose the appropriate answer from the options given below: How many factors of number N is a two digit number? I. N has total number of factors as 16. II. N has 4 single digit prime factors.

Directions: Below question is followed by two statements labelled I and II. Decide if these statements are sufficient to conclusively answer the question. Choose the appropriate answer from the options given below:

How many factors of number N is a two digit number?

I. N has total number of factors as 16.

II. N has 4 single digit prime factors.

A Statement I alone is sufficient to answer the question.

B Statement II alone is sufficient to answer the question.

C Statement I and Statement II together are sufficient, but neither of the two alone is sufficient to answer the question.

D Either Statement I or Statement I alone is sufficient to answer the question.

E Neither Statement I nor Statement II is necessary to answer the question.

Solution

Correct Answer: Option C

From Statement 1

We know that, when a number can be expressed as a₁^m1 × a₂^m2 × a₃^m3 × …. × a_n^mn,

Where, a₁, a₂, a₃… ,an are prime numbers and m₁, m₂, …, m_n are positive integers.

Total number of factors of the number = (m₁+1) × (m₂+1) × (m₃+1) × …(m_n+1)

N has a total of 16 factors.

∴ 16 = (m₁+1) × (m₂+1) × (m₃+1) × …(m_n+1)

Case 1: 16 = 1 × 16 = (0 + 1) × (15 + 1)

N can have just 1 prime factor raised to 15.

Case 2: 16 = 2 × 8 = (1 + 1) × (7 + 1)

N can have 1 prime factor raised to 7 and the other raised to 1.

Case 3: 16 = 2 × 2 × 4 = (1 + 1) × (1 + 1) × (3 + 1)

N can have two prime factors raised to 2 and 1 prime factor raised to 3

Case 4: 16 = 4 × 4 = (3 + 1)× (3 + 1)

N can have just 2 prime factors, each raised to the power of 3

Case 5: 16 = 2 × 2 × 2 × 2 = (1 + 1) × (1 + 1) × (1 + 1) × (1 + 1)

N could have 4 prime factors each raised to the power 1.

Thus, statement 1 alone is insufficient to answer the question.

From Statement 2

The prime factors of N are 2, 3, 5 and 7.

But, we do not know the total number of factors and hence we cannot determine the powers to which these prime factors are raised.

∴ We can determine only a few double-digit factors but not all of them.

Thus, statement 2 is also insufficient to answer the question.

From Statements 1 and 2

N has 16 factors in total and has prime factors 2, 3, 5and 7.

∴ Clearly, all the prime factors are raised to the power 1.

∴ the number of 2- digit factors is 9

Thus, Statement I and Statement II together are sufficient, but neither of the two alone is sufficient to answer the question.

Solution

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