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

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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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23 Nov 2019, 01:42
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.

Hi Bunuel
Is the pattern above similar to "of all the rectangles with a given perimeter, the one with greatest area is a square"?
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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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29 Jan 2020, 03:30
m +p is max when they are equally divided and < 100 so a value that must be equally divided is 98
98/2 =40 x^2 = 49 x=+-7 as x is positive int we can consider x=7 and y=7
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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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22 May 2020, 02:45
Hi!
Can someone pls explain why we cant assume that M+P<10 then? (taking square root from both sides). its is given that both are positive numbers...
I was trying to solve thinking that M+P is at most 9 then, and the biggest products is therefore 20 (4*5).
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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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22 Jun 2020, 23:35
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

Since m and p are positive integers, mp would be integer(int. * int. = another int.). Factoring answer choices into largest possible factors(only two) would help.

Also, as greatest value is asked for, we can check for the condition $$m^2 + p^2 < 100$$ to hold true by checking answer choices in descending order.
51 = 17 * 3; as $$17^2 > 100$$ Not valid
49 = 7 * 7; $$7^2 + 7^2 = 98 < 100$$ Valid

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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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Updated on: 02 Jul 2020, 02:21
m^2 + p^2 < 100
(m - p)^2+2mp < 100

Divide by 2..

mp < 50 - (m - p)^2 / 2

mp is maximum when (m - p)^2 = 0
Then;
mp < 50
maximum mp=49

Originally posted by BlueRocketAsh on 01 Jul 2020, 12:56.
Last edited by BlueRocketAsh on 02 Jul 2020, 02:21, edited 1 time in total.
If m and p are positive integers and m^2 + p^2 < 100, what is the   [#permalink] 01 Jul 2020, 12:56

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