2021 · Q85
Let n (>1) be a composite natural number whose square root is not an integer. Consider the following statements:
- 1n has a factor which is greater than 1 but less than the square root of n.
- 2n has a factor which is greater than the square root of n but less than n.
Which of the statements given above is/are correct?
General Mental Ability, Quantitative Aptitude and StatisticsNumber system and arithmetic
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Since n is composite with an irrational square root, write n = ab with integers 1 < a < b < n. Then a < √n (a cannot equal √n, which is irrational), and correspondingly b > √n. Thus n has a factor strictly between 1 and √n (statement 1) and a factor strictly between √n and n (statement 2) — e.g. n = 12 gives 2 < √12 < 4. Hence both statements are correct, option (C).