Number properties - prime factors

[koby_gen]

If y is an integer such that 2 < y < 100 and if y is also the square of an integer, what is the value of y ?

(1) y has exactly two prime factors.

(2) y is even.

[Anurag@Gurome]

If y is an integer such that 2 < y < 100 and if y is also the square of an integer, what is the value of y ?

(1) y has exactly two prime factors.
(2) y is even.

Statement 1: As y has exactly two prime factors and y is also square of an integer, y must be a square of a prime number. Therefore possible values of y are 4, 9, 25 and 49.

Not sufficient.

Statement 2: Different values of y are possible.

Not sufficient.

1 & 2 Together: Only possible value of y is 4.

Sufficient.

The correct answer is C.

Note: The statement 1 doesn't say "y has exactly two different prime factors". In which case there will be only one possible value of y in that range. Which is 36. And thus statement 1 will be sufficient.

[kevincanspain]

I would say the latter case holds, as 4 does not have two prime factors

[prachich1987]

I would say the latter case holds, as 4 does not have two prime factors

I am not getting above
4 has exactly 2 prime factors
4=2 X 2

Can you please explain your view?

[chendawg]

My thought process was this:

We know 2 < y < 100 and y is also the square of an integer. So we know that y is the square of some number less than 10 and more than 1, so we can consider 2-9.

1. y has exactly two prime factors. Primes under 10 are 2, 3, 5 and 7. Since we know Y is the square of an integer, all its factors must come in 2's. Since Y has 2 prime factors, only (2*3)^2 is under 100. Thus sufficient.

2. y is even. Y can be 2,4, etc. Insufficient.

I got A as the answer.

However, as with Anurag@Gurome's answer, it could be C, as I thought of statement one as saying 2 distinct prime factors, where it just states y has 2 prime factors.

What's OA and source?

[ikaplan]

A cannot be the answer.

Statement (1) states that y has exactly 2 prime factors.

Possible values for y are 4 (2^2), 9(3^2), 16(4^2), 25(5^2) etc...

So there are more than one possible solutions for y from Statement (1), namely 4, 9, 25 etc.

Therefore, A is insufficient

[banibhusan]

Here statement 1 says that there are exactly two prime factors of y. It doesn't mention whether they are distinct or not. So the possible answers statement 1 are 4,9,25 and 49. 36 can be ruled out because it has 2 distinct prime factors, but a total of 4 prime factors which contradicts statement 1. So its insufficient.

Now coming to statement 2, it says that y is even. Which means y can have any even integer between 2 and 100. Hence insufficient.

However if you combine statement 1 and 2, you can eliminate 9,25 and 49 from the possible answers. So the value of y would be 4 which has exactly two prime factors and its even and square of an integer.

Since I am just a beginner, plz correct me if I am wrong.

[ClaireSpencer]

a) is insufficient, b) is sufficient.

Eligible numbers 2<n<100 are 4,9,16,25,36,49,64,81.

1 is not a prime number.

9,25,49,81 have only one prime factor each; 3,5,7,3 respectively.

4 only has one prime factor (2) as does 64 (8^2).

The factors of 36 are 1,2,3,6,18. 2 and 3 are prime numbers.

The answer is 36.

[ClaireSpencer]

I mixed up the information! 1) is sufficient, 2) is insufficient.

[MAAJ]

2 < y < 100
y = x^2 = x * x
y = ?

1) y has exactly 2 prime factors

x * x = y
2 * 2 = 4
3 * 3 = 9
5 * 5 = 25
7 * 7 = 49

2) y is even

x * x = y
2 * 2 = 4
4 * 4 = 16
6 * 6 = 36
8 * 8 = 64

Combining 1) and 2)

Only 2 x 2 = 4

[Ian Stewart]

If y is an integer such that 2 < y < 100 and if y is also the square of an integer, what is the value of y ?

(1) y has exactly two prime factors.
(2) y is even.

Statement 1: As y has exactly two prime factors and y is also square of an integer, y must be a square of a prime number. Therefore possible values of y are 4, 9, 25 and 49.

If I were to ask the question 'how many positive factors does 9 have?', the answer would be 'three': 1, 3 and 9. We would not count the '3' twice. By the same token, if I ask 'how many prime factors does 9 have?' the answer ought to be one; there is no logical reason to count the '3' twice.

When a question asks 'how many prime factors does x have?', that normally means 'how many *different* prime factors does x have?' Fortunately the wording of real GMAT questions will be unambiguous -- GMAT questions will include the phrase 'distinct prime factors' so there is no possibility of misinterpretation. Still, I'm sure Statement 1 in the original post above is intended to mean "y has exactly two *distinct* prime factors" (in which case y must be 36), particularly since the upper bound of 100 seems specifically chosen so that Statement 1 will be sufficient.

[force5]

IMO c

[rohu27]

can we really expect such question on gmat where in its not clearly mentioned if the factors are distinct or not?
i dont think so.

[lfoss]

My take is A
1) Statement 1 is sufficient
with 3 & 2 as Factors

[gmatters12]

I pick A

We know y=integer^2 and y is between 2and 100
So...
y can be 2^2=4
y can be 3^2=9
y can be 4^2=16
y can be 5^2=25
y can be 6^2=36
y can be 7^2=49
y can be 8^2=64
y can be 9^2=81


1) y has exactly 2 prime factors

The only one that has 2 distinct prime factors is 36 (2 and 3 are its only prime factors). All the other numbers have only one prime factor. So statement 1 is sufficient!

2) that does not tell us anything. could be 4, 16, 36, or 64. Insufficient