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If x and y are both integers greater than 1, is xy>100 ?

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If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

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New post 26 Jul 2018, 21:12
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If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

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New post 26 Jul 2018, 23:20
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Bunuel wrote:
If x and y are both integers greater than 1, is xy > 100?

(1) x has exactly 7 unique factors.

(2) y has exactly 9 unique factors.


Given situation:- \(x>1, y>1\) (x and y are integers)
Stem:- Is \(xy>100\)

St1:- x has exactly 7 unique factors
Or, we can write in the prime factorization form,\(x=a^p\), where p+1=7 or, p=6 (I have taken x=a^p not x=a^p*b^q..., because we are given 7 factors ,if we consider more than one prime exponent, we can't get a multiplication result of 7 by multiplying more than one integer values(where none of the integers are 1)) or \({(p+1)(q+1)...}\neq7\)
So,\(x=a^6\)
Now \(minimum(x)=min(a^6)=2^6=64\) (we can't consider a=1 since it yields x=1^6=1)
Therefore, \(x*y=64*2=128>100\) (The lowest integer value of y greater than 1 is 2)
For all other positive integer values of a>2,we have x > 100. Subsequently \(xy>100\).

Sufficient.

St2:-y has exactly 9 unique factors
With the same reasoning as stated in st1, we have \(y=a^p*b^q\), where (p+1)(q+1)=9=3*3.
So, p+1=3 and q+1=3
Or, p=2 and q=2
So, \(min(y)=min(a^2*b^2)=2^2*3^2=36\)
Therefore, \(xy=2*36=72<100\)----------------(a) (when x=2)
and \(xy=3*36=108>100\)-----------------------(b) (when x=3)
From (a) and (b), st2 is insufficient since st2 is inconsistent with the question stem.

Ans. (A)
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If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

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New post 26 Jul 2018, 21:34
Any integer with odd number of unique factors is a perfect square

If x and y are both integers greater than 1

To determine whether xy > 100 ?

Statement 1

x has exactly 7 unique factors.

=> x is a perfect square

=> x \(\geq\) 64 since no perfect squate less than 64 has 7 number of factors

=> Since y is an integer and y > 1 => xy \(\geq\) 128

Statement 1 is sufficient

Statement 2

y has exactly 9 unique factors.

=> y is a perfect square

=> y \(\geq\) 36 since no perfect square less than 36 has 9 factors

=> Since x is an integer and x > 1 => xy \(\geq\) 72

Statement 2 is not sufficient

Hence option A
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Re: If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

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New post 29 Jul 2018, 06:51
Correct me if I'm wrong but doesn't 24 also have 7 factors, making statement 1 insufficient? I am a bit unsure about this. Hence I went ahead with option C.
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Re: If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

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New post 29 Jul 2018, 07:05
nithinjohn wrote:
Correct me if I'm wrong but doesn't 24 also have 7 factors, making statement 1 insufficient? I am a bit unsure about this. Hence I went ahead with option C.


Hi nithinjohn ,

\(24=2^3*3^1\)
So the no of factors=(3+1)*(1+1)=8

And the factors are 1,2,3,4,6,8,12, and 24.

https://gmatclub.com/forum/divisibility ... 74998.html
You may visit above link for more clarity.

FINDING THE NUMBER OF FACTORS OF AN INTEGER
First make 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 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: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

This is available in the link provided.
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Re: If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

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New post 29 Jul 2018, 08:02
Bunuel wrote:
If x and y are both integers greater than 1, is xy > 100?

(1) x has exactly 7 unique factors.

(2) y has exactly 9 unique factors.



xy>100 and both x and y>1. MEANS both are at least 2 and therefore if we can prove at least one is >50, it would be sufficient

1) x has exactly 7 unique factors
    a) 7=1*7, no other possibilities so it is a^6
    b) odd number of factor means X is perfect square
    c) if it were just 3 factors.. it meant a perfect square of prime number
So least value =2^6=64
So minimum value of xy is 2*64=128>50
Sufficient

2) y has 9 unique factors.
    a) higher number of factors necessarily does not mean LARGER number
    b) 9=1*9....so least value 2^8=256>50 ....yes
    c) 9=3*3.... So type a^2*b^2 least value = 2^2*3^2=4*9=36>50....No
So xy>100 and xy<100 possible
Insufficient

A
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: If x and y are both integers greater than 1, is xy>100 ? &nbs [#permalink] 29 Jul 2018, 08:02
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