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# x is an integer and x raised to any odd integer is greater

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x is an integer and x raised to any odd integer is greater [#permalink]  08 Jun 2010, 00:14
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x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity 7^(x-1)-5^x?

(1) z < 25 and w=7^x
(2) x = 4
[Reveal] Spoiler: OA

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Last edited by Bunuel on 20 May 2012, 10:57, edited 3 times in total.
Edited the question and added the OA
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Re: DS problem [#permalink]  08 Jun 2010, 03:58
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ykaiim wrote:
x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity $$7^x^-^1 - 5^x$$?

1) z < 25 and w = 7x
2) x = 4

Note: The above expression is 7^(x-1) -5^x.

I guess statement (1) should be: $$z<25$$ and $$w=7^x$$.

"x is an integer and x raised to any odd integer is greater than zero" means $$x=integer>0$$.
Q: is $$w-z>5(7^{x-1}-5^x)$$?

(1) $$z<25$$ and $$w=7^x$$ --> is $$7^x-z>5(7^{x-1}-5^x)$$? --> is $$7^x-z>5*7^{x-1}-5^{x+1}$$? --> is $$7^x-5*7^{x-1}+5^{x+1}>z$$? --> as the lowest value of $$x$$ is 1, then the lowest value of LHS is when $$x=1$$: $$LHS_{min}=7^x-5*7^{x-1}+5^{x+1}=7-5+25=27$$ --> so the lowest value of LHS is 27, which is more than $$z$$ as $$z<25$$ --> hence $$7^x-5*7^{x-1}+5^{x+1}>z$$ is true. Sufficient.

(2) $$x = 4$$. No info about $$w$$ and $$z$$. Not sufficient.

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Re: Is w - z greater than 5 times the quantity 7^x-1 - 5^x [#permalink]  20 May 2012, 08:33
Could you please reply with the detailed explanation for the above question.
i am confused as to how just Statement I is sufficient to answer the question(A), without knowing the value of X?
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Re: Is w - z greater than 5 times the quantity 7^x-1 - 5^x [#permalink]  20 May 2012, 08:52
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Hello, let me try and explain.

paraphrasing the questions gives us, is w-z > 5.7^(x-1) - 5^(x+1). Further, x is a positive integer as an odd power of x is greater than 0.

taking statement 1, you have the question as 7^x-z > 5.7^(x-1) - 5^(x+1)

Now, for all cases where x is a positive integer, 7^x would be greater than 5.7^(x-1). Next, 5^(x+1) would always be >= 25 as the least value of x can be 1.

So, 7^x > 5.7^x-1
and z < 5^(x+1). Reversing this inequality and adding both the inequalities, we get:

7^x - z > 5.7^x-1 - 5^(x+1),

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Re: Is w - z greater than 5 times the quantity 7^x-1 - 5^x [#permalink]  20 May 2012, 09:21
Expert's post
the question is not simple but if you think a bit abstract and test number you can arrive to the solution.

The FIRST important thing, breaking the problem, is to understand tha our X is positive. Infact, if a number raised to power of 3 is >0 that means X itself is positive because the odd powr maintain the original sign of the number, so X must be positive.

At this point using the exponent rules we have w - z > 5*7^x-1 - 5^x+1 ----> 7^x - 24 (from stem x<25) > 5*7^x-1 - 5^x+1

testing number 1 ------> 7 -24 > 1 - 25 ---> - 17 > -24 this is TRUE. also if you test some value < 25 positive but also negative the result is the same.

1) sufficient

x = 4 -------> w - z > 5*7^3 - 5^5..........at this point you know nothing about w and z.

2) insufficient

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Re: Is w - z greater than 5 times the quantity 7^x-1 - 5^x [#permalink]  03 Jun 2012, 00:27
Bunuel wrote:
carcass wrote:
x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity 7^x-1 - 5^x?

1 z < 25 and w = 7^x
2 x = 4

why can't x be zero apart from even ? any odd number raised to the power zero is 1 which is an interger greater than zero. I think x is given as an interger and not only positive integer. if you work with x = 0 , statement 1 is not sufficient ?
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Re: Is w - z greater than 5 times the quantity 7^x-1 - 5^x [#permalink]  03 Jun 2012, 01:38
Expert's post
Kaps07 wrote:
Bunuel wrote:
carcass wrote:
x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity 7^x-1 - 5^x?

1 z < 25 and w = 7^x
2 x = 4

why can't x be zero apart from even ? any odd number raised to the power zero is 1 which is an interger greater than zero. I think x is given as an interger and not only positive integer. if you work with x = 0 , statement 1 is not sufficient ?

We are told that integer x raised to to any odd integer is greater than zero: $$x^{odd}>0$$. Now, if $$x=0$$ then $$x^{odd}=0^{odd}=0$$ (for odd>0), which violates given condition, so $$x$$ cannot be zero.

Hope it's clear.
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Re: Is w - z greater than 5 times the quantity 7^x-1 - 5^x [#permalink]  03 Jun 2012, 02:02
Thanks, I misunderstood the question. I thought it was odd int raised to the power x. It is clear now.
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Re: DS problem [#permalink]  06 Sep 2012, 03:59
Bunuel wrote:
ykaiim wrote:
x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity $$7^x^-^1 - 5^x$$?

1) z < 25 and w = 7x
2) x = 4

Note: The above expression is 7^(x-1) -5^x.

I guess statement (1) should be: $$z<25$$ and $$w=7^x$$.

"x is an integer and x raised to any odd integer is greater than zero" means $$x=integer>0$$.
Q: is $$w-z>5(7^{x-1}-5^x)$$?

(1) $$z<25$$ and $$w=7^x$$ --> is $$7^x-z>5(7^{x-1}-5^x)$$? --> is $$7^x-z>5*7^{x-1}-5^{x+1}$$? --> is $$7^x-5*7^{x-1}+5^{x+1}>z$$? --> as the lowest value of $$x$$ is 1, then the lowest value of LHS is when $$x=1$$: $$LHS_{min}=7^x-5*7^{x-1}+5^{x+1}=7-5+25=27$$ --> so the lowest value of LHS is 27, which is more than $$z$$ as $$z<25$$ --> hence $$7^x-5*7^{x-1}+5^{x+1}>z$$ is true. Sufficient.

(2) $$x = 4$$. No info about $$w$$ and $$z$$. Not sufficient.

I don't understand the function of statement z<25. If lowest value of LHS is 27 which is more than z, then it doesn't fulfill the condition given?
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Re: DS problem [#permalink]  02 May 2013, 11:45
Bunuel wrote:
ykaiim wrote:
x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity $$7^x^-^1 - 5^x$$?

1) z < 25 and w = 7x
2) x = 4

Note: The above expression is 7^(x-1) -5^x.

I guess statement (1) should be: $$z<25$$ and $$w=7^x$$.

"x is an integer and x raised to any odd integer is greater than zero" means $$x=integer>0$$.
Q: is $$w-z>5(7^{x-1}-5^x)$$?

(1) $$z<25$$ and $$w=7^x$$ --> is $$7^x-z>5(7^{x-1}-5^x)$$? --> is $$7^x-z>5*7^{x-1}-5^{x+1}$$? --> is $$7^x-5*7^{x-1}+5^{x+1}>z$$? --> as the lowest value of $$x$$ is 1, then the lowest value of LHS is when $$x=1$$: $$LHS_{min}=7^x-5*7^{x-1}+5^{x+1}=7-5+25=27$$ --> so the lowest value of LHS is 27, which is more than $$z$$ as $$z<25$$ --> hence $$7^x-5*7^{x-1}+5^{x+1}>z$$ is true. Sufficient.

(2) $$x = 4$$. No info about $$w$$ and $$z$$. Not sufficient.

Please explain why the lowest value of X is 1.
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Re: DS problem [#permalink]  03 May 2013, 03:03
Expert's post
Rajkiranmareedu wrote:
Bunuel wrote:
ykaiim wrote:
x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity $$7^x^-^1 - 5^x$$?

1) z < 25 and w = 7x
2) x = 4

Note: The above expression is 7^(x-1) -5^x.

I guess statement (1) should be: $$z<25$$ and $$w=7^x$$.

"x is an integer and x raised to any odd integer is greater than zero" means $$x=integer>0$$.
Q: is $$w-z>5(7^{x-1}-5^x)$$?

(1) $$z<25$$ and $$w=7^x$$ --> is $$7^x-z>5(7^{x-1}-5^x)$$? --> is $$7^x-z>5*7^{x-1}-5^{x+1}$$? --> is $$7^x-5*7^{x-1}+5^{x+1}>z$$? --> as the lowest value of $$x$$ is 1, then the lowest value of LHS is when $$x=1$$: $$LHS_{min}=7^x-5*7^{x-1}+5^{x+1}=7-5+25=27$$ --> so the lowest value of LHS is 27, which is more than $$z$$ as $$z<25$$ --> hence $$7^x-5*7^{x-1}+5^{x+1}>z$$ is true. Sufficient.

(2) $$x = 4$$. No info about $$w$$ and $$z$$. Not sufficient.

Please explain why the lowest value of X is 1.

See the highlighted part: x is an integer and x raised to any odd integer is greater than zero means that x is a positive integer, thus its lowest possible value is 1.

Hope it's clear.
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Re: x is an integer and x raised to any odd integer is greater [#permalink]  16 Jun 2013, 20:48
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ykaiim wrote:
x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity 7^(x-1)-5^x?

(1) z < 25 and w=7^x
(2) x = 4

Responding to a pm:

What is the meaning of: x is an integer and x raised to any odd integer is greater than zero?
It is just a convoluted way of saying 'x is a positive integer'. Since x raised to any odd power is positive, x must be positive. x can take values 1, 2, 3, 4 and so on.

Question: Is $$w - z > 5*7^{x-1}-5^{x+1}$$

Using statement 1: Is $$7^x - 25 > 5*7^{x-1} - 5^{x+1}$$

Even though we are given that z is less than 25, let's assume it to be 25. In case we can prove that left hand side is greater than right hand side for z = 25, we can prove that left hand side will remain greater than right hand side for any other value of z. If z has a smaller value, say 9, left had side will become even larger (since a smaller number will be subtracted) than the right hand side.

Now note that $$7^x$$ will be greater than $$5*7^{x - 1}$$ since $$7^x$$ can be written as $$7*7^{x - 1}$$ (x is a positive integer).
Also note that minimum value of x is 1 so $$5^{x + 1}$$ will always be equal to or larger than 25 (x is a positive integer).

So comparing the left and right hand side expressions, $$7^x$$ is greater than $$5*7^{x - 1}$$ and 25 is less than or equal to $$5^{x+1}$$. So left hand side must be greater than the right hand side in all cases when x is a positive integer.

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Re: x is an integer and x raised to any odd integer is greater [#permalink]  05 May 2014, 03:24
paraphrasing the questions gives us, is w-z > 5.7^(x-1) - 5^(x+1). Further, x is a positive integer as an odd power of x is greater than 0.

taking statement 1, you have the question as 7^x-z > 5.7^(x-1) - 5^(x+1)

Now, for all cases where x is a positive integer, 7^x would be greater than 5.7^(x-1). Next, 5^(x+1) would always be >= 25 as the least value of x can be 1.

So, 7^x > 5.7^x-1
and z < 5^(x+1). Reversing this inequality and adding both the inequalities, we get:

7^x - z > 5.7^x-1 - 5^(x+1),

tnx for this
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Re: x is an integer and x raised to any odd integer is greater [#permalink]  26 Jun 2015, 01:04
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Re: x is an integer and x raised to any odd integer is greater [#permalink]  18 Jul 2015, 02:24
x raised to any odd integer is greater than 0 --> x = integer >0 --> x>=1

(1) --> w-z > 7^x - 5^2 = 7.7^(x-1) - 5^2 > 5.7^(x-1) - 5^2

in order to w-z> 5.7^(x-1) - 5^(x+1), x+1 >=2 --> x >=1 --> sufficient.

(2) insufficient

=> A.
Re: x is an integer and x raised to any odd integer is greater   [#permalink] 18 Jul 2015, 02:24
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