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

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

Here are some nice number properties to know,

Most numbers have an even amount of factors because factors usually come in pairs. Take 24 = 1 * 24 = 2 * 12 = 3 * 8 = 4 * 6, there are 8 factors or 4 pairs of factors when you list them this way.

The only way to get an odd number of factors is when the number is a square, we get something like 9 = 1 * 9 = 3 * 3 and we can see the 3 appears twice, but we only count it as one factor. Therefore we can quickly check the squares only to find which ones have 3 factors exactly.

1 has only 1 factor, not included.
4 = 1*4 = 2*2, 3 factors.
9 = 1*9 = 3*3, 3 factors.
16 = 1*16=2*8 ... more than 3 factors.
25 = 1*25 = 5*5, 3 factors.
36 has more than 3 factors.
49 = 1*49 = 7*7, 3 factors.

Therefore only 4 squares qualify.

Ans: A

Rule: in order for any Integer to have only 3 (+)Positive Divisors, it must take the following FORM:

N = (p)^2


*where the Variable P = Prime Number/Prime Base

So basically we are looking for a Prime Integer SQUARED that has a Result in between [1 and 50]

2'2 = 4
3'2 = 9
5'2 = 25
7'2 = 49

only 4 Possibilities

-A-

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

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