If m and p are positive integers and m^2 + p^2 < 100, what is the

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
answer 49

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

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.

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

Answer D.

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.

Answer: D

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

Answer:

Cheers,
Brent

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

Please give Kudos if you like this explanation

Maybe somebody can help me out:

If I simplify \(m² + p² < 100\) by taking the square root it gets me \(m + p < 10\)

Why does no answer choice match the simplified version? Am I not allowed to take the square root or where did my thinking go wrong?

Would really appreciate some help. I do not see my mistake.

The point is \(\sqrt{m^2 + p^2}\) does not equal to m + n. Does \(\sqrt{3^2 + 4^2}\) equal to 3 + 4?

I dont understand how m^2 + p^2 = (m - p)^2+2mp

Can someone explain? Thanks!

(m - p)^2 + 2mp = m^2 - 2mp + p^2 + 2mp = m^2 + p^2.

Hope it's clear.

Im so sorry but I still dont understand this! Is there some basic maths principle I am missing here? I know that the difference of 2 squares is a^2-b^2=(a+b)(a-b), but this is different? Where are the minus' coming from?

Can you expand m^2+p^2 in greater detail?

Thanks so much!

The square of the difference of two terms is \((a - b)^2 = (a - b)(a - b) = a^2 - 2ab + b^2\). Now, if you farther subtract 2ab from that you'd get \(a^2 + b^2\). So, \((a - b)^2 +2ab = (a^2 - 2ab + b^2) + 2ab = a^2 + b^2\).

Answer: Option D

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