if a is a positive integer, is there any integer n where (1<n<a) such that a/n is an integer?

Considering Statement 1 alone
1) a is a multiple of a prime number

For example, lets consider the prime number to be 5.
If a is a multiple of prime number 5, a could be 5 or 10 or 15 or ...
which shows that a could be the prime number itself or any of the other multiples.

If a is the prime number itself, then n takes the vale of any integer between 1 and 5
In this case, a/n is not an integer.

However, if a is not the prime number itself, but takes other values 10 or 15 or ...
n can the vale of 5 and in this case a/n will be an integer (eg. a=15 and n=5, a/n = 3)

As both the above cases are opposing and do not give a unique answer, statement 1 by itself is not sufficient.

Considering Statement 2 alone
2) a is a product of 2 integers.

If a (eg. 6) is a product of two integers (eg. 2 and 3), a is not a prime number.
Additionally, the two factors (eg. 2 and 3) of a will lie between 1 and a.

Thus, n can take up the values of these two factors (eg. 2 and 3) of a (eg. 6)

Therefore, a/n will be an integer.

As statement 2 is sufficient by itself, the Answer is Option B
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if a is a positive integer, is there any integer n where (1<n<a) such that a/n is an integer?

This question seems a little ambiguous for me,

1) a is a multiple of a prime number
CASE 1 : a can be 2,3,5,7 -> In this case answer is NO
CASE 2 : a can be 6(3*2) , 14(7*2) , 15 (5*3) -> In this case answer is YES
INSUFFICIENT

2) a is a product of 2 integers.
CASE 1 : a can be 2,3,5,7 -> In this case answer is NO
CASE 2 : a can be 6,10,15-> In this case answer is YES

INSUFFICIENT.

Statement 1 & 2 together : INSUFFICENT
OPTION : E

Statement 1:

A is a multiple of Prime number

We should remember every number is multiple of itself.

So, considering A as 2 not satisfy the condition mentioned in the question.
While, 4 justify the condition.

Hence statement 1 is not sufficient.

Statement 2:

A is a product of 2 integers

Again any integer is a product of 1(which is an integer) and itself.

2=1*2( Don't satisfy the eqn)
4=2*2( Satisfy the eqn)


Hence statement 2 is not sufficient.

Even by combining statement 1 and 2 we will not get any solution.

So answer should be E

Answer is "D".

Statement 1 : a is a multiple of a prime number.
If a is a multiple of any prime number, then that prime number itself is the possible value for n for which \(a/n\) is a integer.
Eg:
- Let a = 8 (multiple of prime number "2")
There exists 2 values for n : 2 and 4. (where one of them, 2, is the prime number itself)
\(a/n\) = 4 and 2 respectively for values of n as 2 and 4.

- Let a = 33 (multiple of prime number "11")
There exists 2 values for n : 11 and 2. (where one of them, 11, is the prime number itself)
\(a/n\) = 3 and 11 respectively for values of n as 11 and 3.

Therefore we can say that, Statement 1 alone is sufficient to answer the question. Statement 1 will return "Always Yes".


Statement 2 : a is a product of 2 integers.
If a is a product of two integers, then at least both of the given integers can act as value for n for which \(a/n\) is a integer.
Eg:
- Let a = 24 (product of 4 and 6)
There exists at least 2 values for n : 4 and 6.
\(a/n\) = 6 and 4 respectively for values of n as 4 and 6.

Therefore we can say that, Statement 2 alone is sufficient to answer the question. Statement 2 will return "Always Yes".

Since, both of the statements alone are sufficient to answer to given question, answer is D (EACH statement ALONE is sufficient to answer the question).

given a is a positive integer, is there any integer n where (1<n<a)
target a/n is an integer
#1
a is a multiple of a prime number
possible value of a can be multiple values of
a= 2*k ; 3*k ; 5*k
a= 4,6,8 ; 6,9,12 ; 10,15,20
with given condition 1<n<a ; we get both yes and no ; insufficient
#2
a is a product of 2 integers
a can be any value of integer such that we would always get yes & no as value of n may or may not be a multiple so getting an integer always is not sufficient ; eg 1<2<6 ; yes ; 1<5<6 ; no ; where n= 2,5 and a ; 6
insufficient
from 1& 2
again with given condition of 1<n<a ; we get yes & no always ; eg 1<2<6 ; yes ; 1<5<6 ; no ; where n= 2,5 and a ; 6
insufficient
OPTION E


if a is a positive integer, is there any integer n where (1<n<a) such that a/n is an integer?
1) a is a multiple of a prime number
2) a is a product of 2 integers.

Yes there are different values of n.

Option D.

Each statement alone is sufficient.

We can simplify the question as follows:
a and n are positive integers. 1<n<a wherein a/n is an integer. This means that n is a multiple of a.

1. a is a multiple of a prime
We identify a few possibilities.
n=2 a=3 no
n=2 a=6 yes
Different answers, insufficient. A and D are out.

2. a=int. x int.
a=1x3 and n= 2 for example
a is not a multiple of n so no

n=2
a=2x2 yes, n is a multiple of a
Different answers, insufficient.

Combining the 2 options still does not give us any further information so the answer is E.

if a is a positive integer, is there any integer n where (1<n<a) such that a/n is an integer?
1) a is a multiple of a prime number
2) a is a product of 2 integers.

Statement 1) let's say a = 15 ( 3*5, multiple of prime no and is a positive integer)
let's say n = 3 ( 1<3<15 and an integer)
a/n = 15/3=5 (integer), hence, YES there is an integer n where (1<n<a) such that a/n is an integer.

let's say a = 5 ( 1*5, product of two integers and is a positive integer)
let's say n = 3 ( 1<3<5 and an integer)
a/n = 5/3=( non integer), hence, NO there is not an integer n where (1<n<a) such that a/n is an integer.
Not Sufficient

Statement 2) let's say a = 15 ( 3*5, product of two integers and is an positive integer)
let's say n = 3 ( 1<3<15 and an integer)
a/n = 15/3=5 (integer), hence, YES there is an integer n where (1<n<a) such that a/n is an integer.

let's say a = 5 ( 1*5, multiple of prime no and is a positive integer)
let's say n = 3 ( 1<3<5 and an integer)
a/n = 5/3=( non integer), hence, NO there is not an integer n where (1<n<a) such that a/n is an integer.
Not Sufficient

Answer E

A=3*4 =12

n can take any value between 1 to 12 as per question.
Given 1<n<12
When n=3 , a/n IS AN Integer --Yes
When n=5,a/n NOT and Integer.--No

Therefore Insufficient



A=3*4 =12

n can take any value between 1 to 12 as per question.
Given 1<n<12
When n=3 , a/n IS AN Integer --Yes
When n=5,a/n NOT and Integer.--No

Therefore Insufficient

On Combining also, we get a YES as well as a NO.
So IMO E.
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