danzig wrote:
Can the positive integer \(k\) be expressed as the product of two integers, each of which is greater than 1?
(1) \(k^2\) has one more positive factor than k.
(2) \(11 < k < 19\)
I don't understand well this explanation of the OE. Please, your help:
The only types of numbers k such that k2 has exactly one more positive factor than k are primes. Prime numbers have two factors and their squares have three. If k had more than two factors, the number of factors would increase by more than 1 when squared. Thus, k must be prime, answering the question.
Note, for a number K the factors are K,1 and 'few other' depending upon whether it is prime or not
for number k^2 - the factors are 1, k and k^2 and 'few other'
The 'few other' factors depend on the fact whether K is divisible by a number or not.
For example:
factors of 5 : 1, 5 and ? - nothing else
factors of 6: 1, 6 and ? - (2,3)
Similarly,
factors of 25: 1,5, 25 and ? - nothing else ( because 5 is not divisible by anything else and hence can not be broken into any other number)
factors of 36: 1,6,36 and ? - (2,3,4,9, 12,18)
Therefore if you notice the pattern only for a prime number, number of factors of k^2 is one more than number of factors for k.
Now with this concept target the question
stem 1 : it shows us k is prime using above mentioned concept. can a prime number be a product of 2 integers each greather than 1? no. So we have a sufficient statement to say No.
stem 2: K could be anything from 12 to 18. So ans would be if k is 12 then yes, but if k is 13 then no. hence insufficient.
Therefore ans A (only statement 1 is sufficient) it is.
Hope it is clear.