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# If the positive integer x has 8 positive factors, what is the value of

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If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:00
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If the positive integer x has 8 positive factors, what is the value of x?

(1) The product of ANY two positive factors of x is even
(2) x has 7 even positive factors

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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:08
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If the positive integer x has 8 positive factors, what is the value of x?

(1) The product of ANY two positive factors of x is even
(2) x has 7 even positive factors

Option 1: if product of any two positive factors is even then atleast 7 factors of X are even. Now even factors like 6 will not be there. because if 2 and 6 are factor, 3 will be a factor. Hence even factors will only constitute powers of 2. So the highest factor will be 128. hence the number will be 128. - Sufficient

Option 2: X has 7 even positive factors. One of the factors will be 1. Hence rest 7 are even. It follows same explanation as above. - Sufficient

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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:12

given x has 8 positive factor, need to find value of x
from1: if factors of x are 1,2,4,6,8,16,32,64 -->we get one answer multiple answers are possible, as long as factor is even
from 2: x has 7 positive even factors, these numbers can be any thing, except, 1 number is odd.

1+2 : we cannot get value of x, as factors can be

3,4,8,16,32,64,128,256 or

1,4,8,16,32,64,128,256 two diff answers so, not suff so, E
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:12
Given x has 8 positive integer factors.

Which means x might be equal to a^7 or a^1 * b^3 etc... where a,b are prime

1) The product of ANY two positive factors of x is even

For any integer 1 is always a factor, and all prime numbers are odd except for 2. So for statement 1 to be true the only value of x will be 2^7 , because with a odd factor the product of factors ( 1* odd factor) will be odd.

Hence 1 is Sufficient.

2) x has 7 even positive factors
1 is the odd factor for any integer. Hence for statement 2 to be true x again has to be only 2^7

Hence 2 is sufficient.

Hence overall answers Option C i.e both 1 and 2 are individually sufficient!
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If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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Updated on: 09 Jul 2019, 08:05
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The question states that positive integer X has 8 positive factors. Thus x>0, and all factors are greater than 0. These factors can be either 1*2^8 or 1*2^3 * 3^4 or anything else. We do not know at this stage. We need to find the value of X. Let us analyze the statements we have:

Statement 1:
The product of any two positive factors of X is even.

This statement means that if we multiply any of the 8 factors of X by another factor, we will get an even number. As an even number is the product of two even numbers, or Even*Odd number, it means that it is impossible that two odd numbers can be multiplied by each other. Thus, there is only 1 odd factor of the number X. And other factors are even. Speaking about even factor, all even numbers apart from 2 can be divided further, e.g. 4=2*2 or 12=2*2*3.
Thus, X = 2^7 * odd factor. If there is only one odd factor, it is 1. As a result, X=1*2^7.

Statement 1 is sufficient.

Statement 2:
X has 7 even positive factors.

We were able to identify this information in the analysis of previous statement, thus the information here is also sufficient to identify the number.

Statement 2 is sufficient.

There is no need to combine both statements together, as each one separately was enough to find X.

P.S. Here, again, instead of looking for all possible factors, I neglected 1 initially and for this reason, answered the question wrong. However, the question was not telling anything about prime factors and for this reason, 1 should have been also counted as a factor.

Originally posted by RusskiyLev on 09 Jul 2019, 07:16.
Last edited by RusskiyLev on 09 Jul 2019, 08:05, edited 1 time in total.
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If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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Updated on: 10 Jul 2019, 07:02
IMO D:
positive integer x has 8 positive factors ; x^7 ; has 7+1 ; 8 factors
#1
The product of ANY two positive factors of x is even
for this value of x in x^7 has to be even so x=2 ; 128 ; sufficient
#2 x has 7 even positive factors
again value of x can be 2sufficient
IMO D

If the positive integer x has 8 positive factors, what is the value of x?

(1) The product of ANY two positive factors of x is even
(2) x has 7 even positive factors

Originally posted by Archit3110 on 09 Jul 2019, 07:17.
Last edited by Archit3110 on 10 Jul 2019, 07:02, edited 1 time in total.
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:20
1
IMO : D

If the positive integer x has 8 positive factors, what is the value of x?

(1) The product of ANY two positive factors of x is even
(2) x has 7 even positive factors

1: The product of any two positive factors is even means that there are no odd factors except 1.
that mean all the rest 7 factors are even, and different, then answer can only be 128.

Because any number below 128 will have less than 8 even factors, and and number above 128 will have more than 8 factors.
A is sufficient.

2:This point says that there are 7 factors that are even, and only odd can be 1, this gives us 128 only .

For this reason the point B is sufficient as well.

Option D wins.
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If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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Updated on: 09 Jul 2019, 21:47
1
If the positive integer x has 8 positive factors, what is the value of x?

(1) The product of ANY two positive factors of x is even
One of the factors of any number is 1. Product of any 2 factors is even => All the remaining 7 are even. This is same as statement 2.

(2) x has 7 even positive factors
Same as statement 1.

If there are 7 positive factors, then they should be 2,2,2,2,2,2,2.
x should be 2^7

Each statement alone is sufficient.

Option D

Posted from my mobile device

Originally posted by prashanths on 09 Jul 2019, 07:21.
Last edited by prashanths on 09 Jul 2019, 21:47, edited 1 time in total.
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If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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Updated on: 09 Jul 2019, 21:32
1
(A) The product of ANY two positive factors of x is even

This means that x can have AT MOST one odd factor. Since 1 is always a factor, 1 must be the only odd factor of X. Therefore X has 7 even factors which means $$X=2^7$$

Sufficient

(2) x has 7 even positive factors

Since X has 7 even positive factors, there is no question of what could be the 8th factor since 1 is always a factor. Therefore $$X=2^7$$

Sufficient

Originally posted by firas92 on 09 Jul 2019, 07:23.
Last edited by firas92 on 09 Jul 2019, 21:32, edited 1 time in total.
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:24
1
D

X has total 8 positive factors as given.

St 1: Product of any two factors is even, so X has to be 2^7 - sufficient
St 2: X has 7 even factors with total 8 factors, again 2^7 satisfies - sufficient
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:24
1
statement1:the product of 1 even *1 odd =even , 1even *1 even =even since the statement states that the product of any 2 numbers is an even number therefore x= even number
statement 2: there are 7 even numbers therefore the same rule applies hence the product of and even number and an odd number =even therefore D
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:27
1
(1) The product of ANY two positive factors of x is even
--> All factors are even except 1
--> x has to be 2^7

Sufficient

(2) x has 7 even positive factors
--> 7 even factors & factor '1'
--> x has to be 2^7

Sufficient

IMO Option D

Pls Hit Kudos if you like the solution
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:28
2
Let x=a^p, where 'a' is prime. hence 'x' has (p+1) nos of factors.
Here , p+1=8 or p=7

question stem:- x=?

st1:- The product of ANY two positive factors of x is even

factor1*factor2=EVEN implies that anyone of the factor must be even.
Hence, 'a' must be the smallest positive even Prime.
So, x=2^7.
Sufficient.

St2:- x has 7 even positive factors
Implies that there are 7 +ve even factors & '1' as the 8th factor.
So 'a' has to be smallest even prime with p=7.
Hence x=2^7
Sufficient

Ans. (D)
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:29
If the positive integer x has 8 positive factors, what is the value of x?

(1) The product of ANY two positive factors of x is even

So every factor besides one is even.

the other 7 factors thus could be any even no.

Insufficient

(2) x has 7 even positive factors

Just says the same thing

B is out

1/2

Both are the same. C is out

Answer is E. E for kudEs!
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:30
1
If the positive integer x has 8 positive factors, what is the value of x?

(1) The product of ANY two positive factors of x is even
(2) x has 7 even positive factors

X wil have 1 , X as two of the factors. From A: if product of two factors is even then each factor has to be even in th form of 2^n (we already have 1 as odd factor )
hence it would be 1,2,4,8, 16,32,64,128
from B if x has 7 even positive factors again it would be same hence both statements alone are sufficient.
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If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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Updated on: 09 Jul 2019, 08:10
1
(1) The product of ANY two positive factors of x is even - Sufficient, it has to be 2^7 (no. of factors when prime factorized will be a multiple of powers+1 of each prime factor) - only then will the total number of factors be 8 while simultaneously have the product of any factor will be even.
(2) x has 7 even positive factors - Sufficient, When total factors is 8 among which 7 has to be even then again only possible value will be 2^7(1 will be a factor for every number).

IMO D

Originally posted by Lampard42 on 09 Jul 2019, 07:32.
Last edited by Lampard42 on 09 Jul 2019, 08:10, edited 1 time in total.
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:35
1
If the positive integer x has 8 positive factors, what is the value of x?

(1) The product of ANY two positive factors of x is even
1st factor is always 1.
To ensure that product of any 2 is even, all 7 other factors MUST BE EVEN so when multipled by 1 give an even product.

1 2 4 8 16 32 64 128..... here x = 128, and there are exactly 8 factors and product of any 2 is even.

This is the only value of x(128) where these conditions are satisfied. If we take any other number with total 8 factors, there will be an odd factor.

(1) IS SUFFICIENT.

(2) x has 7 even positive factors
In (1) above we already concluded that when we have 7 out of 8 even factors, only possible value of x is 128.

(2) is also SUFFICIENT

Answer: D - Each is sufficient
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:40
1. x has 8 factors and product of any two is even, so 7 out of 8 must be even. This doesn't tell us anything else which can be used to get value of x. Not Sufficient.
2. Again states the same which we deduced in 1. Not sufficient.

Together also these statements are not sufficient.
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Re: If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:41
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If the positive integer x has 8 positive factors, what is the value of  [#permalink]

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09 Jul 2019, 07:45
1
If the positive integer x has 8 positive factors, what is the value of x?
If x has 8 positive factors, x can be of the form $$p_1*p_2^3$$ or of the form $$p^7$$

(1) The product of ANY two positive factors of x is even
If product of ANY 2 factors of x is even then 2 is included in the product, i.e. except 1 all factors are even. 1 is that factor.
$$=> x = 2^7$$
SUFFICIENT
(2) x has 7 even positive factors
x has 7 even positive factor and 1 is other factor
$$x=2^7$$
SUFFICIENT

Either statement alone is SUFFICIENT.

IMO D
If the positive integer x has 8 positive factors, what is the value of   [#permalink] 09 Jul 2019, 07:45

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