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Power Prep Number Properties Woes :(

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Power Prep Number Properties Woes :(

by shilotilo » Fri Jun 20, 2008 4:16 am
Hi,
I have a problem with each of the following problems that came up in the Power Prep:

1) If If r lies between 0 and 1 and s lies between 1 and 2, which of the following must be less than 1?
I r/s
II rs
III s-r
My answer is I and II, while the answer given is "only I". How come?

2)Is there a fast way to do this one ( a POE kind of thing?). It took me a while trying to figure out the units digits of which 3 squares can add upto a unit digit of 5, etc, etc ----
The number 75 can be written as the sum of the squares of 3 different integers. What is the sum of these 3 integers?
A) 17, B)16, C)15, D) 14, E)13

And I have no clue how to go about this one ----
3) n is a multiple of 5 and n=(p^2) q. p and q are prime numbers. Which must be a multiple of 25?
A) p^2 B) q^2 C) pq D) (p^2)(q^2) E) (p^3)q


Can someone please help? Thanks.

Shilo
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by g_beatthegmat » Fri Jun 20, 2008 4:33 am
1) If If r lies between 0 and 1 and s lies between 1 and 2, which of the following must be less than 1?
I r/s
II rs
III s-r
My answer is I and II, while the answer given is "only I". How come?


Plug in number for this. Question says r is in between 0 and 1 and s is in between 1 and 2. Say r = 0.9 and s is 1.9

I) r/s = 0.9/1.9 CORRECT, less than 1
II) rs = 0.9 * 1.9 > 1 Not less than 1
III) s-r = 1.9 - 0.9 = 1 Not less than 1

Thus, I only.

2)Is there a fast way to do this one ( a POE kind of thing?). It took me a while trying to figure out the units digits of which 3 squares can add upto a unit digit of 5, etc, etc ----
The number 75 can be written as the sum of the squares of 3 different integers. What is the sum of these 3 integers?
A) 17, B)16, C)15, D) 14, E)13
See https://www.beatthegmat.com/gmat-prep-t9319.html
And I have no clue how to go about this one ----
3) n is a multiple of 5 and n=(p^2) q. p and q are prime numbers. Which must be a multiple of 25?
A) p^2 B) q^2 C) pq D) (p^2)(q^2) E) (p^3)q
See https://www.beatthegmat.com/gmat-prep-multiples-and-primes-t10432.html
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by shilotilo » Fri Jun 20, 2008 5:10 am
Hi g_beatthegmat,
Thank you for your prompt reply!

However, I still have a question about the first problem. I thought a "must be true" question is the one in which the the answer always has to meet the given criterion.
I understand that 1.9*0.9 is greater than 1. But if I try 1.5*0.5, I will get an answer less than 1. In fact, this "less than 1" thingy seems to appear for a whole lot of values when I multiply a proper fraction (fractions between 0 and 1) with any other number.
Is there a paradox here somewhere or am I missing something?
Help!

Shilo
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by shilotilo » Fri Jun 20, 2008 7:08 am
Anybody????? :(
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by AleksandrM » Fri Jun 20, 2008 9:20 am
I found it useful to pick numbers for this one.

0 < r < 1 and 1 < s < 2

I. r/s try 1/2 divided by 3/2 = 2/6, which is <1

try 8/9 divided by 11/10 = 80/99, which is <1

You notice that because the number in the denominator is more than 1, when flipped, it always gives you a number less than 1.

II that's easy: 3/2 x 1/2 = 3/4, but 3/2 x 8/9 = 24/18.

III again 3/2 - 1/2 = 1.

So, you are left with just I.

I'll be honest, took me a little under 2 minutes, whereas it could have probably taken someone else under a minute to solve this using logic.

But, we each do what we can.
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by Ian Stewart » Fri Jun 20, 2008 9:39 am
shilotilo wrote:Hi g_beatthegmat,
Thank you for your prompt reply!

However, I still have a question about the first problem. I thought a "must be true" question is the one in which the the answer always has to meet the given criterion.
I understand that 1.9*0.9 is greater than 1. But if I try 1.5*0.5, I will get an answer less than 1.

Shilo
All you're demonstrating above, by choosing r = 0.5 and s = 1.5 is that rs might be less than 1. That doesn't mean it must be less than 1, which is what the question asks. If you can find a single example where 0<r<1, 1<s<2, and rs is greater than 1, certainly it is not true that rs must be less than 1- you've just proven that it sometimes is not. It might be less than 1, but it might not be. I don't think I understand what you're finding paradoxical here?
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by shilotilo » Fri Jun 20, 2008 8:49 pm
Hi Ian,
Thank you for your reply. Now that you've explained it so clearly, there is no paradox.

Shilo
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by ksh » Sat Jun 21, 2008 3:28 am
hi shilotilo,

For your sl.no. 2 & 3 we can get as below:
2. 2^2+5^2+6^2=75
so, sum of nos. would be 2+5+6=13

3. if n is divisible by 5 and n=p^2q then for n to be divisible by 25 it should be p^3q

Hope u get it.
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by shilotilo » Sat Jun 21, 2008 8:23 am
Hi krsh,
Thank you for your reply.

I solved Q.2. My query was whether there was a quick way to know which squares we were looking for in the first place.

About Q3, The answer is p^2q^2. The solution is in the link in one of the posts above.

Shilo
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by beeparoo » Sat Jun 21, 2008 3:28 pm
@ksh: You made an error in your calculations.
ksh wrote:2. 2^2+5^2+6^2=75
so, sum of nos. would be 2+5+6=13
2^2+5^2+6^2 does not equal 75.
However, you coincidentally got the right answer (i.e. 13) by accident...

As for Question 3 and why you are also incorrect with that one, someone posted a good link above to the correct solution.
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by Thouraya » Sat Jun 19, 2010 4:39 am
75= 5^2 x 3
So Im assuming the sum of the integers is 5+5+3, but I am not sure how these are "different"? theyre only 2 different not 3..
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