4 votes 4 votes Consider numbers greater than one that satisfy the following properties: They have no repeated prime factors; For all primes $p \geq 2$, $p$ divides the number if and only if $p − 1$ divides the number. The number of such numbers is $0$ $5$ $100$ Infinite None of the above Quantitative Aptitude tifr2014 quantitative-aptitude difficult numerical-computation + – makhdoom ghaya asked Nov 9, 2015 • edited Mar 29, 2021 by soujanyareddy13 makhdoom ghaya 1.2k views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
Best answer 6 votes 6 votes The prime factors of $30$ are $2,3,5$, so it satisfies the 1st constraint. However, $\matrix{ (2-1) & \rm divides & 30 & \color{green}\checkmark\\ (3-1) & \rm divides & 30 & \color{green}\checkmark\\ (5-1) & \rm divides & 30 & \color{red}\times\\ }$ and thus it doesn't satisfy the 2nd constraint. One can prove that for $n$ to satisfy these properties, $n=p(p−1)$ for some prime $p$ and that $(p−1)$ satisfies the properties too. Some examples of the numbers that satisfy both constraints are: $$\begin{align} &2\\ 2 \times 3 = &6\\ 6 \times 7 = &42\\ 42 \times 43 = &1806 \end{align}$$Now $1807$ is not a prime and hence breaks the sequence. So, number of such numbers is $4.$ Correct Answer: E. Pragy Agarwal answered Nov 9, 2015 • selected Jun 9, 2019 by Arjun Pragy Agarwal comment Share Follow See all 7 Comments See all 7 7 Comments reply Show 4 previous comments Pragy Agarwal commented Nov 10, 2015 reply Follow Share Yes, kinda lengthy. One can prove that for $n$ to satisfy these properties, $n = p (p-1)$ for some prime $p$ and that $(p-1)$ satisfies the properties too. Once that is achieved, it is a simple matter of noticing that $1807$ is not a prime and hence breaks the sequence. 2 votes 2 votes srestha commented Nov 10, 2015 i edited by srestha Nov 10, 2015 reply Follow Share okay, It is going to check longest chain ,right? But upto how many times should I check , as there is no upper limit ? because there will be always a chance of error in result. Am I right sir? 0 votes 0 votes Arjun commented Nov 10, 2015 reply Follow Share Well, answer is 4. And the reason is told by Pragy. This is the only chain possible and it goes from 2, 6, 42. 1806. After this 3263442 comes and it doesn't satisfy the second constraint and hence we got the answer. But to try this for first time in exam is tricky. I guess for these questions smart people should get 0 and not negative :) 6 votes 6 votes Please log in or register to add a comment.
0 votes 0 votes (a) no repeated prime number.So it can be{2,3,5,6(2*3 so no repeatation of prime in 6),7,11, 13,15,..........,} (b) it can be (2,3)=6 (6,7)=42 ............... (10,11)=110 So, we can say, as prime no is infinite, answer will be (d)infinite srestha answered Nov 9, 2015 • edited Nov 10, 2015 by srestha srestha comment Share Follow See all 0 reply Please log in or register to add a comment.