What is the number of elements of this interesting set ?
Number Theory
Level
1
If A is the set of prime factors of a number k and if B is the set of prime factors of the number (k+1) and if C is defined as AnB ( A intersection B), then What is n(C) ?. (Number of elements of the set C). Kindly exclude counting 1 as a prime factor.
The answer is 0.
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1 solution
If a number "k" is divisible by 2 the next higher number that is divisible by 2 is (k +2). Similarly, if a number k is divisible by a prime number p the next higher number divisible by p is k +p. If the prime factors of a number are p1,p2,p3 ....pn where p1 < p2 < p3..<pn it follows that p1 >= 2 and all the other prime factors are greater than 2. So, the smallest number greater than k which will have a common prime factor is ( k + 2). Therefore ( k +1) and k will not have any common prime factors other than 1. The set AnB is hence a null set. It follows then that n(AnC) =0