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M27-08

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New post 16 Sep 2014, 01:27
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The product of three distinct positive integers is equal to the square of the largest of the three numbers, what is the product of the two smaller numbers?


(1) The average (arithmetic mean) of the three numbers is \(\frac{34}{3}\).

(2) The largest number of the three distinct numbers is 24.

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New post 16 Sep 2014, 01:27
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Official Solution:


This is a 700 question.

Let the three integers be \(a\), \(b\), and \(c\), where \(0 \lt a \lt b \lt c\). Given: \(abc=c^2\) so, \(ab=c\). Question: \(ab=c=?\)

(1) The average (arithmetic mean) of the three numbers is \(\frac{34}{3}\). This basically means that \(a+b+c=34\). Substitute the value of \(c\) with \(ab\) to get \(a+b+ab=34\). From that we can write \((a+1)(b+1)=35\). Now, since \(a\) and \(b\) are integers, then \(a+1=5\) and \(b+1=7\). \(a=4\) and \(b=6\), hence \(ab=24\). Sufficient. (Notice that \(a+1=1\) and \(b+1=35\) is not possible since in this case \(a=0\) and we are told that all integers are positive).

(2) The largest number of the three distinct numbers is 24. Directly gives the value of \(c\). Sufficient.


Answer: D
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Re: M27-08  [#permalink]

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New post 10 Aug 2015, 18:37
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Bunuel,

How did you get from a+b+ab=34 to (a+1)(b+1)=35?

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Re: M27-08  [#permalink]

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New post 23 Aug 2015, 04:41
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Hello,

a + b + ab = 34

add +1 to both sides

a + b+ ab + 1 =34+1

a(1+b) + (b+1) = 35

(b+1) (a + 1) =35

Hope it helps.


Press kudos please if it helps!
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New post 26 Apr 2018, 02:22
I didn't get the part in which you go from:

a(1+b) + b + 1 = 35

to

(a+1)(b+1) = 35

Could you please elaborate just a lil more?


Thanks a lot!

cheers to y'aal!
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New post 26 Apr 2018, 02:31
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New post 26 Apr 2018, 02:33
Obviously.... silly me!

Thanks
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Re: M27-08  [#permalink]

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New post 18 Sep 2018, 07:39
Quote:
(2) The largest number of the three distinct numbers is 24. Directly gives the value of cc. Sufficient.


With #2 alone, c*x = 576, which means a*b = 24, there are two solutions with distinct positive integers.

a = 3, b=8, c=24
a*b*c = 3*8*24 = 576 = 24^2

a = 4, b=6, c=24
a*b*c = 4*6*24 = 576 = 24^2

Using the information from 1), we know that a=3, b=8 is not correct. However, I don't see how we would know this from 2) alone.

Please explain.
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New post 18 Sep 2018, 09:04
I also have the same query as Double down.kindly explain.Or is it because we are asked only abt ab?

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New post 18 Sep 2018, 09:07
monicasharma365 wrote:
I also have the same query as Double down.kindly explain.Or is it because we are asked only abt ab?

Posted from my mobile device



Yes.

The product of three distinct positive integers is equal to the square of the largest of the three numbers, what is the product of the two smaller numbers?
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Collection of Questions:
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M27-08  [#permalink]

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New post 18 Sep 2018, 10:27
the question stem says three distinct positive integers and the product is equal to the square of the highest no.
which means the product is equal to the highest number.

lets say abc, c being the highest no.
a*b*c= c^2
which implies ab=c

first choice explanation
a+b+c=34
adding 1 on both sides
a+b+c+1=34+1
we know ab=c
a+b+ab+1=35
a+ab+b+=35
a(1+b) + (1+b)=35
taking (1+b) out
(1+b)(1+a)=35
factors of 35=1,5,7,35
(1,35) cannot be the pair as a or b is a positive integer
(5,7) is the pair
which means (4,6) is a pair for (a,b)

hence the product is 24
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