Can the positive integer p be expressed as the product of two integers

SOLUTION

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

The wording makes this question harder than it is actually.

If the positive integer p cannot be expressed as the product of two integers, each of which is greater than 1, then it simply means that p is a prime number. So, basically question asks whether p is a prime number.

(1) 31 < p < 37 --> between these numbers there is no prime. Hence ANY integer from these range CAN be expresses as the product of two numbers. Sufficient.

(2) p is odd --> odd numbers can be primes as well as non-primes. Not sufficient.

Answer: A.
_________________

It seems that you misinterpreted the question.

Look at the definition of a prime number: a prime number is a positive integer with exactly two factors: 1 and itself. Now, the questions asks: "can the positive integer p be expressed as the product of two integers, each of which is greater than 1" So, the question basically asks whether p is a prime number, because if it is then p can NOT be expressed as the product of two integers, each of which is greater than 1.

(1) states: 31 < p < 37. Between these numbers there is no prime. Hence ANY integer from these range CAN be expresses as the product of two numbers, which means that the answer to the question is YES: p can always be expressed as the product of two integers, each of which is greater than 1. Sufficient.

Just to illustrate: 32=2*18, 33=3*11, 34=2*17, 35=5*7, 36=2*18.

Hope it's clear.
_________________

I don't understand this question. I am getting E.

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

Statement (1) states that 31<p<37

My logic was that p can be anything that fits between 31 and 37, so 36
36 = 9 x 4 and each integer is greater than 1, so it is sufficient
36 = -9 x -4 and each integer is not greater than 1, so insufficient.

Statement (2), I agree it is insufficient.

(1) and (2) is still insufficient because 35 (fits statement 2) and 35 can be 7x5 so each integer is greater than 1. 35 can also be -7x-5 so each integer is not greater than 1.

Am I misunderstanding the question?

Can the positive integer P be expressed as a product of two integers,each of which is greater than 1?
(1) 31<p<37
(2) p=odd

The answer according my program is A, but I dont understand why it can not be D. Because if we take 3*3=9 which is odd and integer and greater than 1?

Thank you in advance

Merging similar topics.



P is some particular integer and we are asked whether it can be expressed as a product of two integers, each of which is greater than 1. Now, for (2) if p=9 then the answer is YES, it can be expressed as a product of two integers, each of which is greater than 1 but of p=5 then the answer is NO, it cannot be expressed as a product of two integers, each of which is greater than 1. Two different answers, hence this statement is not sufficient.

Does it makes sense?

P.S. Please refer for a complete solution to the above posta and ask if anything remains unclear.
_________________

Thank you very much Bunuel,

So basically if there are 2 possible answers (yes/no) it will always be insufficient?

It's a YES/NO DS question. In a Yes/No Data Sufficiency question, statement is sufficient if the answer is “always yes” or “always no” while a statement is insufficient if the answer is "sometimes yes" and "sometimes no".
_________________

Hi Bunuel,
The question asks: can p be expressed as a product of two numbers greater than 1? If I prove that p can be expressed as a product of two numbers which are < 1 and can also be expressed as a product of two numbers > 1 then the statement A) / B) would become insufficient. Right?

Now, as quoted above, for statement A), I can express 36= 1*36, -1*-36,-9*-4..... so doesn't it mean that statement A) is insufficient?

If A) is sufficient to answer the question, is it because of the fact that the question asks "Can it be expressed as product of two numbers > 1" instead of "Is P a product of two numbers which are always greater than 1?"

No, the red part is not correct.

The question asks "can p be expressed as the product of two integers, each of which is greater than 1".

If from a statement you get that EACH possible value of p can be expressed as the product of two integers, each of which is greater than 1, then the answer is YES, and the statement is sufficient.

If from a statement you get that NONE of the possible values of p can be expressed as the product of two integers, each of which is greater than 1, then the answer is NO, and the statement is sufficient too.

If from a statement you get that some possible values of p cannot but other possible values of p can be expressed as the product of two integers, each of which is greater than 1, then we'd have two answers to the question and the statement wouldn't be sufficient.

Hope it's clear.
_________________

What if i say that P = 33 x 1.

So it implies that surely one of the integer is greater than 1 but other one is 1 itself ?

Mental block..!!!

33 also can be expressed as 3*11, so the answer to the question "can the positive integer P be expressed as a product of two integers, each of which is greater than 1?" is still YES.
_________________

I agree, but then I came across this question.

Can the positive integer n be written as the sum of two different positive prime numbers?

(1) n is greater than 3.
(2) n is odd.

and I am confused again.

In both questions we know that the number is an integer,
let's suppose the number is 33 for both questions(for easier calculation)
33 can be 33 x 1 or 11 x 3. (can be true)
& 33 can be 31 + 2 or 27 + 6 (can be true) than why is the answer E in this particular case ??


O_o confused. Please help.

For original question, EVERY possible value of p (32, 33, 34, 35, and 36) CAN be written as the product of two integers, each of which is greater than 1. Thus answer B.

For the other question, SOME possible values of n (for example n=5) CAN be written as the sum of two different positive prime numbers and others cannot (for example n=11). Thus answer E.

The second question is discussed here: can-the-positive-integer-n-be-written-as-the-sum-of-two-126725.html

Hope it helps.
_________________

condition 1 - number that falls in given range are - 32 (2*16), 33 (3*11), 34 (2*17), 35 (7*5), 36 (2*18) .Since , each of number can be expressed as product of two integer which are greater than 1 so answer is YES.

condition 2 - odd integer - 3 ->NO
15 ->YES (3*5)

Since , only condition 1 is sufficient to answer given question and condition 2 does not answer the question so answer is (A)

Required: Can p be expressed as the product of two integers, each of which is greater than 1
Or is p = x*y, where x and y are greater than 1.
This means p can have the numbers that are not prime, since a prime number has only 2 factors: 1 and the number itself.

Statement 1: 31 < p < 37
Values of p can be = 32, 33, 34, 35, 36
None of these is prime, hence p can be written as a product of x and y
SUFFICIENT

Statement 2: p is odd.
Odd numbers can both be prime and non prime
INSUFFICIENT

Correct Option: A

Target question: Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?

This question is a great candidate for rephrasing the target question.
If an integer p can be expressed as the product of two integers, each of which is greater than 1, then that integer is a composite number (as opposed to a prime number). So . . . .

REPHRASED target question: Is integer p a composite number?

Aside: We have a video with tips on rephrasing the target question (below)

Statement 1: 31 < p < 37
There are 5 several values of p that meet this condition. Let's check them all.
p=32, which means p is a composite number
p=33, which means p is a composite number
p=34, which means p is a composite number
p=35, which means p is a composite number
p=36, which means p is a composite number
Since the answer to the REPHRASED target question is the SAME ("yes, p IS a composite number") for every possible value of p, statement 1 is SUFFICIENT

Statement 2: p is odd
There are several possible values of p that meet this condition. Here are two:
Case a: p = 3 in which case p is not a composite number
Case b: p = 9 in which case p is a composite number
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Answer : A

RELATED VIDEO

Need to find if p is prime or not.
As non prime numbers can be expressed as product of two factors.

Between 31 and 37 :
These are the numbers
32,33,34,35,36 : All of these numbers are not prime. So it can be expressed as product of two factors.

p is odd.
All odd numbers are not always prime.

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

(1) 31 < p < 37

None of these numbers are prime. Therefore, we can express each number in the range as the product of two integers. SUFFICIENT.

(2) p is odd.

81 = yes.
11 = no.

INSUFFICIENT.

Answer is A.
_________________

Solution:

Question Stem Analysis:

We need to determine whether integer p can be expressed as the product of two integers, each of which is greater than 1. That is, we need to determine whether p is a composite number.

Statement One Alone:

We see that p can be 32, 33, 34, 35, or 36. Since each one of these numbers is a composite number, statement one alone is sufficient.

Statement Two Alone:

Statement two alone is not sufficient. For example, if p = 15, then p can be expressed as the product of two integers, each of which is greater than 1 (notice that 15 = 3 x 5). However,if p = 17, then p can’t be expressed as the product of two integers, each of which is greater than 1 (notice that 17 is a prime).

Answer: A
_________________