If x is a positive integer, is x prime?

Question

(1) x has the same number of factors as y^2, where y is a positive integer greater than 2.

(2) x has the same number of factors as z, where z is a positive integer greater than 2.

Bunuel · Accepted Answer

If x is a positive integer, is x prime?

(1) x has the same number of factors as y^2, where y is a positive integer greater than 2.

y^2 is a perfect square. The number of distinct factors of a positive perfect square is ALWAYS ODD, while the number of factors of a prime is two (1 and itself). Thus since x has the same number of factors as a perfect square it cannot be a prime. Sufficient.

(2) x has the same number of factors as z, where z is a positive integer greater than 2. Clearly insufficient.

Answer: A.

FN · Answer

I will go with A..

all prime factors have even number of factors i.e 2

any square of a positive integer will give you odd factors..sufficient..x is not a prime.

pmenon · Answer

said A for this one.

y^2 = 9,16,25,36 and so on.

All these numbers have more than two factors, and so x cannot be prime.

B says z=3,4,5,6 and so on. Some of the numbers are prime, and some are not, so this statement by itself is insufficient.

thorinoakenshield · Answer

Hi there,

Can you please explain the statement "The number of distinct factors of a positive perfect square is ALWAYS ODD"? e.g. distinct factors of 36 are 2 and 3 (=even). Likewise for 100 are 2 and 5 ( = even).

Thank you,

TO

Bunuel · Answer

It says distinct factors, not distinct  prime  factors.

So, for example, distinct factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36: 9 factors. 
Distinct factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100: 9 factors.
Distinct factors of 4 are 1, 2, and 4: 3 factors.

Does this make sense?

vivapopo · Answer

1) SUFF.: If x has the same number of factors as y2, then x cannot be prime. A prime number is a number that has only itself and 1 as factors. But a square has at least 3 prime factors. For example, if y is prime, y = 2, then y2 = 4, which has 1, 2, and 4 as factors. If the root (in this case y) is not prime, then the square will have more than 3 factors. For example, if y = 4, then y2 = 16, which has 1, 2, 4, 8, and 16 as factors. In either case, x will have at least 3 factors, establishing it as nonprime.

(2) INSUFF.: If z is prime, then x will have only two factors, making it prime. But if z is nonprime, it will have either one (if z = 1) or more than two factors, which means x will have either one or more than two factors, making x nonprime. Since we do not know which case we have, we cannot tell whether x is prime

bumpbot · Answer

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If x is a positive integer, is x prime?

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If x is a positive integer, is x prime?

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