(Number) n(A) denotes the number of positive divisors of a positive integer A.

[Max@Math Revolution]

(Number) n(A) denotes the number of positive divisors of a positive integer A. How many A’s are there satisfying n(A) = 3 between 1 and 50, inclusive?

A. 4
B. 10
C. 15
D. 17
E. 25

[Max@Math Revolution]

Solution:

Divisors or factors of a number, say M, are the integers that can divide into M without a remainder. That is, if we have a number M, such that M = ab and a and b are positive integers, then a and b are factors of M.

When M is expressed as a product of its prime factors only, then we say that we have prime factorized M. If we prime factorize a positive integer, M, as M = \(p_1^ {t_1}\) * \(p_2^ {t_2}\) * ……* \(p_n ^ {t_n}\), where \(p_i\) stands for different prime numbers, and \(t_i\) are positive integers and stands for the exponents of the different prime factors or divisors, then the number of factors of M = (\(t_1\) + 1) * (\(t_2\) + 1)......(\(t_n\) + 1).

The important part here is the word “different.”

In the question, n(A) denotes the number of positive divisors of a natural number A. We are required to find the total number of A’s that satisfy n(A) = 3 between 1 and 50, inclusive.

To have 3 positive divisors, A must have a single prime factor with the highest power of 2. This is possible when prime numbers are squared.

=> \(2^2\) = 4 → 3 divisors → 1, 2, and 4.

=> \(3^2\) = 9 → 3 divisors → 1, 3, and 9.

=> \(5^2\) = 25 → 3 divisors → 1, 5, and 25.

=> \(7^2\) = 49 → 3 divisors → 1, 7, and 49.



Hence, there are 4 A’s that satisfy n(A) = 3 between 1 and 50, inclusive.

Therefore, A is the correct answer

Answer A

[Scott@TargetTestPrep]

(Number) n(A) denotes the number of positive divisors of a positive integer A. How many A’s are there satisfying n(A) = 3 between 1 and 50, inclusive?

A. 4
B. 10
C. 15
D. 17
E. 25

Solution:

Recall that if p is a prime, then p^2 has exactly 3 positive divisors: 1, p, and p^2. Therefore, A = p^2 where p is prime, and we need to determine the number of numbers between 1 and 50 (inclusive) that have this property. Since 2^2, 3^2, 5^2 and 7^2 are the only numbers between 1 and 50 (inclusive) that have this property, the correct answer is 4.

Answer: A