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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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26 Jul 2018, 21:12
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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.

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

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

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

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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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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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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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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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