If a and b are positive integers, is a/b an integer?

stat 1: not suff

stat 2: not suff

considered together,,, both stats not suff, hence E

Hi everyone!

If a and b are positive integers, is a/b an integer?

(1) a>b
(2) a and b share only one factor.

we have this question. I got it wrong because I said B (answer E)

I said B because is says the numbers are primes and if you divide any two primes you will never get an integer. UNLESS the primes are the same number.

So here is my question: The question stem says that A AND B are + integers. I didn't test the a=b possibility because the stem says "A and B", so clearly two different numbers...

Or.. am I wrong on this?

In the math language, if I say I have two variables A and B, am I saying that those numbers MUST be different? I mean it would make sense to assume it right? Or I must say that I have two distinct variables A and B?

Does it have to be clearly stated in the stem that the numbers are "distinct, positive integers, bla bla" for me to discard the a=b possibility? I get this doubt really often! Should I always check the a=b scenario if the stem doesn't state that the numbers are different?

Do I make myself clear here?

Thank you for your input

cheers!

Am I wrong to say that from 2) we get that the numbers are prime? Is 1 included in the possible pool of numbers?

If two positive integers share only 1 positive factors, we see those two numbers are "relatively prime."
However, this does not necessarily mean the two numbers are prime.
For example, 8 and 25 share only one factor (1)
Likewise, 30 and 49 share only one factor (1)


Yes, 1 is a possible value.
For example, 1 and 1 share only one factor (1)
And 24 and 1 share only one factor (1)

Cheers,
Brent

Target question: Is a/b an integer?

Given: a and b are positive integers

Statement 1: a > b
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of a and b that satisfy statement 1. Here are two:
Case a: a = 2 and b = 1, in which case a/b = 2/1 = 2, and 2 IS an integer
Case b: a = 3 and b = 2, in which case a/b = 3/2, and 3/2 is NOT an integer
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: a and b share only one factor
There are several values of a and b that satisfy statement 2. Here are two:
Case a: a = 2 and b = 1, in which case a/b = 2/1 = 2, and 2 IS an integer
Case b: a = 3 and b = 2, in which case a/b = 3/2, and 3/2 is NOT an integer
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several values of a and b that satisfy BOTH statements. Here are two:
Case a: a = 2 and b = 1, in which case a/b = 2/1 = 2, and 2 IS an integer
Case b: a = 3 and b = 2, in which case a/b = 3/2, and 3/2 is NOT an integer
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

For a/b to be integer, a must be multiple of b.

In statement 1, if a>b, it is not necessary that a is a multiple of b.
In statement 2, if a and b share only one factor, they are co-prime. Here a will be multiple of b only if b is 1. Since it is not given that b is necessarily 1, data is insufficient.

By combining both also, it is not necessary that a will be a multiple of b.

Hence, together also insufficient.

Thanks,
Anant Saraogi

1. Unless it is explicitly stated otherwise, different variables CAN represent the same number.

2. a and b share only one factor does not mean that a and b are primes. Consider a = 6 and b = 35.

Thank you so much for your answers! Got it now :D

E. Both the statements are insufficient.

1. a > b

There are multiple possibilities for a & b and is thus insufficient.

2. a and b share only one factor

Let a=15 & b=10. They share one factor 5. But this does not give us a/b as an integer(1.5).

Thus this is also not sufficient.

Combining statement 1 & 2, with the above example, we can conclude that this is also not sufficient.


Thus, the answer is E.

Official solution from Veritas Prep.

Solution: E.

In order for a/b to be an integer, then b must be a factor of a. Statement 1 tells us that a is greater than be, which is necessary but not sufficient for a/b to be an integer. Statement 2 tells us that our variables have only one factor in common; we know this factor is 1. This may appear to tell us that a/b can never be an integer, but don’t forget to consider b=1. If that is the case, then a/b is always an integer. If b is any other positive integer, then a/b is never an integer. Insufficient. When combined, we still run into the situations outlined for Statement 2, so we still cannot solve. The correct answer is E.

Good question...

Given: +int a,b
Question: is a/b = int? Or... is a>b and which is an E/O number?

(1) a>b
If a=2,b=1 then 2/1 = 2, YES
If a=3,b=2 then 3/2 = 1.5, NO
Insufficient.

(2) a and b share only one factor.
If a=1, b=1 then 1/1 = 1, YES
If a=2, b=4 then 2/4 = 0.5, NO
Insufficient.

(3)
As above, if a=3, b=2 then 3/2 = 1.5, NO.
Insufficient.

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