Can the positive integer k be expressed as the product of

Can the positive integer k be expressed as the product of two integers, each of which is greater than 1?

Question basically asks whether k is a prime number. If it is, then it cannot be expressed as the product of two integers, each of which is greater than 1 (definition of a prime number).

(1) k^2 has one more positive factor than k --> if k is a prime then it has 2 factors: 1, and k --> k^2 will have one more, so 3 factors: 1, k, and k^2. If k is some composite number greater than 1, then it has more than 2 factors and # of factors of k^2 will increase by more than just by 1 (try any composite number to check this). If k=1 then k^2 will have the same # of factor as k: one. Hence k=prime. Sufficient.

(2) 11 < k < 19. k can be 13, so prime, as well as 14 so not a prime. Not sufficient.

Answer: A.

Hope it's clear.
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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.

Statement 1: K^2 has one more factor. Let's say if K was a 4 (non-prime ) then it will have 4 factors. [1,2,4]. If you square 4, 16 has 5[1,2,4,8,16] factors. The factors increased by 2. The only way for a factor to increase by 1 is if the number is prime. 13 has 2 factors [1,13]. 169 has 3 factors [1,13,169].

This mean K is a prime. The only 2 numbers that can multiple to 13 is 13 and 1. Therefore we know the answer for the question is NO.

Answer A is sufficient

I think the question lacks a bit of information. It mentions that both the integers should be greater than 1 but does not mention that both integers should be different. However, still, when prime numbers are squared, the number of different factors increases by 1 which is a product of the prime no multiplied by the prime number.

Can the positive integer k be expressed as the product of two integers, each of which is greater than 1?

Question basically asks whether k is a prime number. If it is, then it cannot be expressed as the product of two integers, each of which is greater than 1 (definition of a prime number).

Bunuel - I take it you mean product of two DIFFERENT integers, each of which is greater than 1.

If k is some composite number greater than 1, then it has more than 2 factors --> So K will be prime or not? Am I reading something wrong Bunuel?

A prime number is a natural number with exactly two distinct natural number divisors: 1 and itself. Otherwise a number is called a composite number.

So, composite numbers are not primes.

We are considering 3 cases for (1):
k=prime;
k=composite>1;
K=1;

And get that k can be only a prime number.



It doesn't really matter. Can you express a prime as the product of two same integers?
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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.

Basically, what the question asks is whether k is not a prime number.

1)k is prime. Sufficient.
2)k can be 12,13,14,15,16,17,18. Both primes and non primes appear in this set. Insufficient

Answer is A.

Kudos Please... If my post helped.

Merging similar topics. Please refer to the solutions above.
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K may or may not be prime. Who said anything about the factor other than k being not equal to 1?

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