Subject
Quantitative Aptitude
Topic
Data Sufficiency
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.
ক)
Statement I alone is sufficient to answer the question.খ)
Statement II alone is sufficient to answer the question.গ)
Statement I and Statement II together are sufficient, but neither of the two alone is sufficient to answer the question.ঘ)
Either Statement I or Statement I alone is sufficient to answer the question.ঙ)
Neither Statement I nor Statement II is necessary to answer the question.Explanation
From Statement 1 We know that, when a number can be expressed as a 1 m1 × a 2 m2 × a 3 m3 × …. × a n mn , Where, a 1 , a 2 , a 3 … ,an are prime numbers and m 1 , m 2 , …, m n are positive integers. Total number of factors of the number = (m 1 +1) × (m 2 +1) × (m 3 +1) × …(m n +1) N has a total of 16 factors. ∴ 16 = (m 1 +1) × (m 2 +1) × (m 3 +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.Related questions
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