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Director  B
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If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

Originally posted by AbdurRakib on 17 Jun 2017, 08:20.
Last edited by Bunuel on 17 Jun 2017, 13:26, edited 2 times in total.
Renamed the topic and edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 60515
If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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1
36
AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

There is a property saying that for given sum of two numbers, their product is maximized when they are equal.

Thus mp will be maximzed if given that $$m^2 + p^2 < 100$$ when m=p. In this case we'd have:

$$2m^2 < 100$$
$$m^2 < 50$$
And since given that m is a positive integer then m = 7.

mp = 7*7 = 49.

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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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My solution
m^2 + p^2 = (m - p)^2+2mp<100, so if we assume that (m - p)^2 = 0 in order to maximize the value of 2mp, then 2mp < 100, mp < 50
##### General Discussion
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If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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AbdurRakib wrote:
If m and p are positive integers and $$\sqrt{m} + \sqrt{p}$$ < 100, what is the greatest possible value of mp ?
A. 36
B. 42
C. 48
D. 49
E. 51

AbdurRakib
Kindly change the highlighted part of the question. it must be $$m^2 + p^2 < 100$$
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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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3
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Given m and p are postive integers and we need to find out the greatest value of mp?

Since $$m^2 + p^2$$ < 100, evaluating the answer options..
36 = 6*6, 42 = 6*7, 48 = 6*8, 49 = 7*7 and 51 = 3*17

Of the available only 51(3*17) has a value of $$m^2 + p^2$$ greater than 100
Since we have to find out the greatest value of mp,
49, where m=p=7 (Option D) will have value of $$m^2 + p^2$$ =$$7^2 + 7^2$$ = 98 which is less than 100
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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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Bunuel wrote:
AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

There is a propert saying that for given sum of two numbers, their product is maximized when they are equal.

Thus mp will be maximzed if given that $$m^2 + p^2 < 100$$ when m=p. In this case we'd have:

$$2m^2 < 100$$
$$m^2 < 50$$
And since given that m is a positive integer then m = 7.

mp = 7*7 = 49.

http://gmatclub.com/forum/if-x-2-y-2-10 ... 47064.html
http://gmatclub.com/forum/if-the-popula ... 30222.html
http://gmatclub.com/forum/the-lengths-o ... 28319.html
http://gmatclub.com/forum/what-is-the-g ... 91398.html
http://gmatclub.com/forum/m05-183677.html
http://gmatclub.com/forum/given-that-ab ... 27051.html
https://gmatclub.com/forum/if-x-and-y-a ... 99904.html

Hope it helps.
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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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Does this mean that in GMAT questions we can always assume x may be equal to y, unless specifically mentioned that they are 'distinct numbers'?
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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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ameyaprabhu wrote:
Does this mean that in GMAT questions we can always assume x may be equal to y, unless specifically mentioned that they are 'distinct numbers'?

Unless it is explicitly stated otherwise, different variables CAN represent the same number.
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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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2
AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

AM >= GM
$$(m^2+p^2 )/ 2 >= \sqrt{m^2*p^2}$$
$$mp =< (m^2+p^2 )/ 2 <100/2$$
mp < 50

Greatest possible value = 49

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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

The product of two positive integers is greatest if they are as close as possible, that is, if they are equal. Thus, we can let p = m, and our inequality becomes m^2 + m^2 < 100. Let’s solve it:

2m^2 < 100

m^2 < 50

m < √50

Since m is a positive integer and the largest positive integer less than √50 is 7, m = 7. In that case, p is also 7. Thus the greatest possible value of mp is 7 x 7 = 49.

Alternate Solution:

Let’s test each answer choice, starting from the greatest, which is 51.

Notice that 51 = 3 x 17, so our only choices for m and p are 3 and 17 or 1 and 51. Neither of these choices satisfy m^2 + p^2 < 100, and therefore mp cannot equal 51.

Next, let’s test 49. Since the choice m = 49 and p = 1 does not satisfy m^2 + p^2 < 100, let’s take a look at m = 7 and p = 7. Since m^2 + p^2 = 49 + 49 = 98 < 100, mp can equal 49. Since we are looking for the greatest possible value of mp, it is 49.

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Director  P
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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

I think gmat will not force us to remember any rules here. just pick specific numbers and see what happen

if m=p, we have m=7
increase m and reduce p
m=8, p=5
m=9, p=3
so, it is best if m=p
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Posts: 4217
Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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Top Contributor
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AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

Let's test some pairs of values.
Since m and p are POSITIVE INTEGERS, we won't have a ton of options

Try m = 9 and p = 4 (aside: if m = 9, then 4 is the biggest possible value of p)
In this case, mp = (9)(4) = 36

Try m = 8 and p = 5 (aside: if m = 8, then 5 is the biggest possible value of p)
In this case, mp = (8)(5) = 40

Try m = 7 and p = 7 (aside: if m = 7, then 7 is the biggest possible value of p)
In this case, mp = (7)(7) = 49

Try m = 6 and p = 7 (aside: if m = 6, then 7 is the biggest possible value of p)
In this case, mp = (6)(7) = 42

Try m = 5 and p = 8
At this point, we can see that, if we continue, we'll be duplicating the work we did earlier.
That is, this case (m = 5 and p = 8) is the SAME as the 2nd case we examined.
If we continue, the next case we test will be m = 4 and p = 9. which is the SAME as the 1st case we examined, etc.

Since we've now tested all possible (and relevant) cases, we can see that the maximum value of mp is 49

Cheers,
Brent
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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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1
1
AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

I learned one of these concepts from Bunnel

($$(M-P)^2>= 0$$
Therefore M^2 +P^2 >= 2MP . Greatest Value of 2MP = M^2+P^2 . Great Value of M^2+P^2 = 99 or 98
There MP = 99/2 = Integer (not possible - Integer X Integer = Integer)
or 98/2 =49

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Posts: 1
Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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I think the answer should be 42. The question says m and p are integers. 'Are integers' should mean two distinct integers, not a single integer. So, we cant take 7 as the value of both m and p.
Rather we can take 6 and 7 where the sum of their square is 85 that is lower than 100. And 7 times 6 equals to 42.
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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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Jewelvigilant wrote:
I think the answer should be 42. The question says m and p are integers. 'Are integers' should mean two distinct integers, not a single integer. So, we cant take 7 as the value of both m and p.
Rather we can take 6 and 7 where the sum of their square is 85 that is lower than 100. And 7 times 6 equals to 42.

You are wrong. Unless it is explicitly stated otherwise, different variables CAN represent the same number. Please re-read the discussion above.
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If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

Using Bunnel and Satsurfs explanations above, here is how i made sense of the solution:
M^2+P^2 <100
If (m-p)^2 =0
then m^2 +p^2-2mp=0; m^2+p^2 =2mp
Inserting this into the original equation;
Therefore; 2mp < 100; mp<50
The greatest possible value of mp is therefore 49
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Joined: 15 Oct 2016
Posts: 28
If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

The basic concept behind nailing this problem is the fact that for real positive numbers a and b, (a+b)/2 >= (ab)^0.5. This is known as the AM-GM inequality.

Therefore, (m^2+p^2)/2 >= mp => mp <50. Hence the greatest value of mp is 49 (without the restriction on m and p of being integers).

Since we have got this for any real positive m and p, you just need to ensure 49 is possible as a product of integers (since m and p have to be integers). If no, then keep moving backwards from 49.

Luckily, 49 does get expressed as a product of two integers. Hence, max (mp) = 49
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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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meyba2ty wrote:
AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

Using Bunnel and Satsurfs explanations above, here is how i made sense of the solution:
M^2+P^2 <100
If (m-p)^2 =0
then m^2 +p^2-2mp=0; m^2+p^2 =2mp
Inserting this into the original equation;
Therefore; 2mp < 100; mp<50
The greatest possible value of mp is therefore 49

The expression above gives you the max product of real positives m and p (not necessarily for integers).

So, please make sure you do not miss the part where you need to check whether the maximum product value of 49 is valid for product of integers.
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If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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another approach factorize the options
option c----- 49
(7,7) 49+49= 98<100 hold to this value
when you check other options 98 will be the highest
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Re: If m and p are positive integers and m^2 + p^2 < 100, what is the  [#permalink]

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Bunuel wrote:
AbdurRakib wrote:
If m and p are positive integers and $$m^2 + p^2 < 100$$, what is the greatest possible value of mp ?

A. 36
B. 42
C. 48
D. 49
E. 51

There is a property saying that for given sum of two numbers, their product is maximized when they are equal.

Thus mp will be maximzed if given that $$m^2 + p^2 < 100$$ when m=p. In this case we'd have:

$$2m^2 < 100$$
$$m^2 < 50$$
And since given that m is a positive integer then m = 7.

mp = 7*7 = 49.

You note that "for a given sum of two numbers, their product is maximized when they are equal." Is this also true for a "given sum of squares of two numbers?" The question, mentions $$m^2 + p^2 < 100$$, not $$m + p < 100$$. Re: If m and p are positive integers and m^2 + p^2 < 100, what is the   [#permalink] 13 Jan 2019, 10:05

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