If p and s are different prime numbers, how many different positive

Question

If p and s are different prime numbers, how many different positive factors does ps^2 have? A. Two B. Three C. Five D. Six E. Seven 2024-01-29_19-22-01.png

Bunuel · Accepted Answer

Since p and s are different prime numbers,  ps^2  will have (1 + 1)(2 + 1) = 6 positive factors: 1, p, s, ps, s^2, ps^2.

Answer: D.

P.S  Finding the Number of Factors of an Integer 
 
First, make the prime factorization of an integer  n = a^p * b^q * c^r , where  a ,  b , and  c  are prime factors of  n , and  p ,  q , and  r  are their respective powers.
 
The number of factors of  n  will be expressed by the formula  (p+1)(q+1)(r+1) .  NOTE:  this will include 1 and  n  itself.
 
 Example:  Finding the number of all factors of 450:  450 = 2^1 * 3^2 * 5^2 
 
The total number of factors of 450, including 1 and 450 itself, is  (1+1)(2+1)(2+1) = 2*3*3 = 18  factors.

Priyankjain5 · Answer

Bunuel  IF i take 3 and 5 as the two prime numbers then 5*3^2 = 225 , which has 5*5*3*3 ,     = 9 factors which is not in the option

Bunuel · Answer

It's p*s^2, not (ps)^2.

GmatKnightTutor · Answer

Plugging in two different prime numbers could be helpful.

ps^2

2 (3^2)

2 times 3 times 3 = 18

18 has how many different POSITIVE factors?

1 x 18
2 x 9
3 x 6

We have a total of 6 different positive factors.

(D) is your answer.

emont · Answer

can we use this same formula for both questions "how many positive factors" and "how many factors" ?

Bunuel · Answer

If you want to include negative divisors, you simply multiply the number of positive divisors (which you get from the formula) by 2, since each positive divisor has a corresponding negative divisor.

slabarca · Answer

How wrong am I in my reasoning, if I assumed that "how many  different  positive factors" meant that we only were looking for non repeated different factors?

Let's say prime numbers 3 and 5 = 3 * 5 * 5 = 75. Then factors are= 75, 1, 25, 5, 5 and 3 -- Different factors = 5, Total factors = 6

Can someone help me out with that so I don't get it wrong again?

Bunuel · Answer

75 = 3 * 5^2

The number of positive factors of 75 is (1 + 1)(2 + 1) = 6. These factors are 1, 3, 5, 15, 25, and 75.

The number of positive factors and the number of distinct positive factors are the same. So, 75 has six positive factors, which is the same as saying it has six distinct positive factors.

Similarly, the number of prime factors and the number of distinct prime factors is also the same. For example, 75 has two prime factors, 3 and 5, which is the same as saying it has two distinct prime factors, 3 and 5.

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If p and s are different prime numbers, how many different positive

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If p and s are different prime numbers, how many different positive

Prep Toolkit

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