Latest Elitmus Aptitude Question SOLUTION: Is x has only '3' factors? a) x^2 has only '5' factors. b)one of its factor must be prime number.

only first condition is sufficient as x=4 has 3 factors and 16 has 5 factors.
using statement 2 only it is nt possible to tell whether it ha s 3 factors or not?

on the basis of people answer over here
lets first take x= 4 factors = 1,2,4
x^2 = 16 factors = 1,2,4,8,16
as per this both condition is true

we take x= 9 factors = 1,3,9
x^2 = 81 factors = 1,3,9,27,81

so over here also both statements conditions are sufficient

consider s1: x^2 has 5 factors means the possible prime factorization for x^2 is a^4 where a is the prime factor of x^2 and (a^2)^2 here a^2 is the x so x has 3 factors sufficient and s2: is not sufficient bcz when a number has more than 2 factors one of its factor is prime...so s2:is not sufficient so option A:s1 is sufficient and s2 alone is not sufficient

b is sufficient for this question

both conditions are sufficient for the value of x=4

none of the condition satisy the equation

both equations are insufficient as 4 and 9 both are possibile

(A)- only statement (a) is sufficient.
Only squares of primes has 3 factors. For example, 4 has three factors. 1, 2, 4.
Now when we square 4, we get 16, which has 5 factors. 1, 2, 4, 8, 16. So Statement 1 is sufficient.
Statement two says that one of the factor must be prime number. 10 has 4 factors. i.e., 1, 2, 5, 10 and one of which is prime. but we cannot say whether numbers of this format always have three factors or not. So statement 2 is insufficient.

Question is not asking about value of x so x can be 4 or 9 whatever.
does it has 3 factors or not?

Both the statement will be used to answer

any one plz suggest which is correct solution?

This is a question about prime numbers in disguise. Of course, any prime number has exactly two factors: 1 and itself. If we multiple two different primes, say 2 and 5, we get a number with four factors: the factors of 10 are {1, 2, 5, 10}. The only way to get a number with exactly three factors is if the number is the perfect square of a prime number. For example, 9 is the square of 3, and the factors of 9 are {1, 3, 9}; 25 is the square of 5, and the factors of 25 are {1, 5, 25}. If P is a prime number, then the factors of P squared are (a) 1, (b) P, and (c) P squared. Three factors. That’s what the question is asking. Is N the perfect square of a prime number.

Statement #1:

If N is a prime number itself, then its square only has three factors. Squaring a prime doesn’t produce enough factors. This doesn’t meet the condition of this statement.

If N is the product of two primes, then its square has 7 factors. For example, 2*5 = 10, and 10 square is 100 which has seven factors: {1, 2, 4, 5, 10, 20, 25, 100}. Squaring a product of primes produces too many factors. This also doesn’t meet the condition of this statement.

The only way the square of N could have five factors is if N is the square of a prime number. Suppose N = 2 square, which is 4. Then N squared would be 16, which has five factors: {1, 2, 4, 8, 16}. In general, if P is a prime number, and N equals P squared, then N squared would equal P to the 4th power, which has five factors:

Therefore, N must be the square of a prime number, so we can give a clear “yes” answer to the prompt question. This statement, alone and by itself, is sufficient.

Statement #2:

This is tricky. If we know that N is the square of a prime number, then this statement would be true, but that’s backwards logic. We want to know: if this statement is true, does it allow us to conclude that N is the square of a prime number? If only factor of N is a prime number, then N could be:

(a) a prime number: the only prime factor of 7 is 7.

(b) any power of a prime number: the only prime factor of 7 to the 20th is 7

So N could be the square of a prime number, or just the prime number itself, or the cube of the prime number, or the fourth power, or etc. Any number of factors would be possible, so we have no way to answer the question. This statement, alone and by itself, is not sufficient.

Answer = (A)