How many different even positive factors does the integer P have? (1)

Question

How many different even positive factors does the integer P have?

(1) P =  x^2 y^2 z^3 2^4 , where x, y, and z are different odd prime numbers.

(2) Total number of different positive factors of P is 180. P is a multiple of 16 but not of 32. The only other prime numbers that are factors of P are 3, 5 and 7.

chetan2u · Accepted Answer

How many different even positive factors does the integer P have?
 x^2y^3z ..You can find the positive factors by (2+1)(3+1)(1+1)=3*4*2=24
For finding even factors, you have to find the odd factors first by neglecting the power of 2..
So say if y is 2, then odd factors are taken from x^2z ..(2+1)(1+1)=3*2=6 , and the remaining 24-6 are even factors.

So here we require to know the if even prime is a factor of P and what all other prime factor are there.

(1) P =  x^2 y^2 z^3 2^4 , where x, y, and z are different odd prime numbers.
Here the odd factors become (2+1)(2+1)(3+1)=3*3*4=36
Total factors= (2+1)(2+1)(3+1)(4+1)=180
Even factors = 180-36
Sufficient

(2) Total number of different positive factors of P is 180. P is a multiple of 16 but not of 32. The only other prime numbers that are factors of P are 3, 5 and 7.
Total factors =180..
Multiple of 16 but not of 32 means it has 2^4..
Thus (odd factors)(4+1)=180..odd factors = 180/5=36..
Even factors =180-36
Sufficient

D

TarunTilokani · Answer

chetan2u   Bunuel   VeritasKarishma  any tips on how to solve this question?

anmolsharma14 · Answer

How can we segregate odd and even positive factors ?

Does it only Depends upon prime numbers in the Integer ?

Karmesh · Answer

We need to find the number of EVEN positive factors. Theory- Any number multiplied by 2 gives an even number. Let's dive in. STATEMENT 1: You are given the prime factors of P as well as their powers. Also, it is stated that x,y,z are different odd prime numbers. This means that x, y & z are prime numbers greater than 2 (e.g 3, 5, 7....). (This is precisely what we need to find the number of factors) Keep in mind you don't really need to find the #factors, just knowing that you can is good enough. STATEMENT 2: This statement essentially restates statement 1. Number of factors is given to be 180. Which is the same as (2+1)(2+1)(3+1)(4+1). " P is a multiple of 16 but not 32" essentially means 2^4 is a factor of P. " The only other prime numbers that are factors of P are 3, 5 and 7" - "x, y, and z are different odd prime numbers" (as clarified above) Hence, D Hope this helped

Karmesh · Answer

Yes, it depends only on the prime numbers. After every other number is just a combination of prime numbers. "COMPOSITE" number or in layman terms "composed of". If a factor is not multiplied by 2 it is an odd factor and vice versa. For e.g in the current question. 3, 9, 5, etc are all odd factors. Whereas, 6, 10, 18, 36, etc are all even factors). Let me know if you need more clarity

KarishmaB · Answer

Check out these two posts too. A lot of these concepts will become clear. https://anaprep.com/number-properties-f ... -a-number/ https://anaprep.com/number-properties-r ... e-factors/ https://anaprep.com/number-properties-f ... ct-square/ and these videos: https://youtu.be/DxIH8rjhpKY https://youtu.be/Kd-4cH4cqHw

MSIMBA · Answer

How many different even positive factors does the integer P have?
x2y3zx2y3z..You can find the positive factors by (2+1)(3+1)(1+1)=3*4*2=24
For finding even factors, you have to find the odd factors first by neglecting the power of 2..
So say if y is 2, then odd factors are taken from x^2z ..(2+1)(1+1)=3*2=6 , and the remaining 24-6 are even factors.

So here we require to know the if even prime is a factor of P and what all other prime factor are there.

(1) P = x2y2z324x2y2z324, where x, y, and z are different odd prime numbers.
Here the odd factors become (2+1)(2+1)(3+1)=3*3*4=36
Total factors= (2+1)(2+1)(3+1)(4+1)=180
Even factors = 180-36
Sufficient

(2) Total number of different positive factors of P is 180. P is a multiple of 16 but not of 32. The only other prime numbers that are factors of P are 3, 5 and 7.
Total factors =180..
Multiple of 16 but not of 32 means it has 2^4..
Thus (odd factors)(4+1)=180..odd factors = 180/5=36..
Even factors =180-36
Sufficient

D

IanStewart · Answer

If a number has a prime factorization of (2^4)(odd primes), then for every odd divisor the number has, you can create 4 even divisors, by multiplying the odd divisor by 2, 2^2, 2^3 and 2^4. In other words, if a number is equal to (2^4)(odd primes), then 4/5 of its divisors are odd.

Each Statement tells us our number has a prime factorization that looks like (2^4)(odd primes), and each Statement lets us work out how many divisors the number has, so each Statement is sufficient alone. And you don't need any of the information in the last sentence of Statement 2 to answer the question, so I'm not sure why it's even there.

Aannie · Answer

Given P is an integer.

Considering statement 1:
P = x2 y2 z3 24, where x, y, and z are different odd prime numbers.

As we know if an integer, k = Ma × Nb × Oc × 2d, where M , N and O are different prime numbers, then number of positive factors = (a + 1)(b + 1)(c + 1)(d + 1)
and Number of odd positive factors = (a + 1)(b + 1)(c + 1)

⇒ Number of odd positive factors of P = (2 + 1)(2 + 1)(3 + 1) = 36
and total number of positive factors of P = (2 + 1)(2 + 1)(3 + 1)(4 + 1) = 180

⇒ Number of different even positive factors of P = 180 − 36 = 144

Since we are getting a definite answer from above statement , statement 1 itself is sufficient to provide the answer.

Considering statement 2:
The total number of different factors in P are 180 and P is a multiple of 16 but not a multiple of 32. Also only other prime numbers that are factors of P are 3, 5 and 7.

As P is a multiple of 16 but not 32 => Highest power of 2 present in P is 4.

From above :-
P = 24 × 3a × 5b × 7c
⇒ Number of positive factors of P = (4 + 1)(a + 1)(b + 1)(c + 1), which is given as 180.
⇒ (a + 1)(b + 1)(c + 1) = 180/5 = 36, which is same as number of positive odd factors.
Hence, number of positive even factors = 180 − 36 = 144.

Since we are getting a definite answer from above statement , statement 2 also itself is sufficient to provide the answer.

As statements 1 and 2 both individually are sufficient to provide the answer

Answer: D.

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How many different even positive factors does the integer P have? (1)

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