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If product of integers a and b is negative and sum of their squares is

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If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 15 May 2019, 02:07
00:00
A
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C
D
E

Difficulty:

  75% (hard)

Question Stats:

44% (01:59) correct 56% (02:11) wrong based on 108 sessions

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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 15 May 2019, 02:13
a x b < 0 i.e either a or b is -ve.

0<a^2 + b^2<86

The square neartest to 86 is 81 (9^2).

If one of the variable is 9 then other one must be smallest and 1.

Possible combinations are 9,-1 or -9,1 eitherway their product is -9.

IMO D.
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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 15 May 2019, 02:19
Hi why not 7 & 6
Sum of the squares is 49 + 36 = 85 which is less than 86
Product is 42

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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 15 May 2019, 03:59
EgmatQuantExpert wrote:
If product of integers a and b is negative and sum of their squares is greater than 0 but less than 86, then what is the maximum value of ab?

    A. -42
    B. -36
    C. -18
    D. -9
    E. -1


If a = -9 and b = 1 ----- a^2 + b^2 = 81+1 = 82

ab = -9*1 = -9

If a = -9 and b = 2 ------ a^2 + b^2 = 81+4 = 85

ab = -9*2 = -18

If a = -7 and b = 6 ------ a^2 + b^2 = 49 + 36 = 85

ab = -7*6 = -42

Finding it difficult to understand what the question is exactly asking?

If it is asking in terms of absolute value then correct answer will be -42.

In general terms, possible maximum value should be -9.

Let us wait for the OA.
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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 15 May 2019, 04:13
A IMO - Work Backwards and sign matter little here when testing everything since we square each a and b - Start from max numbers for -36 = 6*-6; 36+36=72; Try max numbers for 42 = 7*-6; 49+36 = 85, just about fits
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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 15 May 2019, 04:31
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chaitanya2402 wrote:
a x b < 0 i.e either a or b is -ve.

0<a^2 + b^2<86

The square neartest to 86 is 81 (9^2).

If one of the variable is 9 then other one must be smallest and 1.

Possible combinations are 9,-1 or -9,1 eitherway their product is -9.

IMO D.

If product of integers a and b is negative and sum of their squares is greater than 0 but less than 86, then what is the maximum value of ab?

Hey what if I consider two integers 1 & -1.
0<1^2 + (-1)^2<86
ab= -1

-1 is greater than -9, hence maximum value.
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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 15 May 2019, 04:37
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LeoN88 wrote:
chaitanya2402 wrote:
a x b < 0 i.e either a or b is -ve.

0<a^2 + b^2<86

The square neartest to 86 is 81 (9^2).

If one of the variable is 9 then other one must be smallest and 1.

Possible combinations are 9,-1 or -9,1 eitherway their product is -9.

IMO D.

If product of integers a and b is negative and sum of their squares is greater than 0 but less than 86, then what is the maximum value of ab?

Hey what if I consider two integers 1 & -1.
0<1^2 + (-1)^2<86
ab= -1

-1 is greater than -9, hence maximum value.


Yes, possible. I thought along the same line, but still wary of the language as it confuses me.
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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 15 May 2019, 08:05
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EgmatQuantExpert wrote:
If product of integers a and b is negative and sum of their squares is greater than 0 but less than 86, then what is the maximum value of ab?

    A. -42
    B. -36
    C. -18
    D. -9
    E. -1

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We need to find the maximum value of AB and not A & B individually or the maximum sum of their squares,

-1 can be written as 1 X -1,

(1)^2 + (-1)^2 =0

Satisfies both the conditions and is the maximum value.
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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 15 May 2019, 08:21
LeoN88 wrote:
chaitanya2402 wrote:
a x b < 0 i.e either a or b is -ve.

0<a^2 + b^2<86

The square neartest to 86 is 81 (9^2).

If one of the variable is 9 then other one must be smallest and 1.

Possible combinations are 9,-1 or -9,1 eitherway their product is -9.

IMO D.

If product of integers a and b is negative and sum of their squares is greater than 0 but less than 86, then what is the maximum value of ab?

Hey what if I consider two integers 1 & -1.
0<1^2 + (-1)^2<86
ab= -1

-1 is greater than -9, hence maximum value.


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If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 15 May 2019, 10:51
given
a*b=-ve ; so either of a or b is -ve integer and
0<a^2+b^2<86

use answer options to solve
    A. -42 ; -2*21 ; not possible to big value of a^2+b^2
    B. -36 ; can be -2*18,-4*9,-6*6 ; again a^2+b^2 is to big
    C. -18; can be -2*9; -3*6 or 1*-18,; a^2+b^2 large value >86
    D. -9 ; can be -1*9; -3*3 ; yes seems a good contender
    E. -1; -1*1 ; yes best of the lot ; lowest value and a^2+b^2 = 2 sufficient


IMO E ; -1

EgmatQuantExpert wrote:
If product of integers a and b is negative and sum of their squares is greater than 0 but less than 86, then what is the maximum value of ab?

    A. -42
    B. -36
    C. -18
    D. -9
    E. -1

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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 16 May 2019, 10:46
chaitanya2402 wrote:
a x b < 0 i.e either a or b is -ve.

0<a^2 + b^2<86

The square neartest to 86 is 81 (9^2).

If one of the variable is 9 then other one must be smallest and 1.

Possible combinations are 9,-1 or -9,1 eitherway their product is -9.

IMO D.



Hi! Really nice approach. That's what I started out with, but then I thought if we included fractions:
e.g.
a= 1/4 and b= -4
Then ab= -1 and 0<a^2 +b^2< 86
(1/16) +16 would still be less than 86
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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 17 May 2019, 23:01
It is C, -18, Product of -9 and 2, Hit and trail.
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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 18 May 2019, 02:26
Please can you clarify the difference between Value and Number? Does the question not ask for the 'value' (rather than 'number') and the sign should therefore be ignored?
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If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post Updated on: 20 May 2019, 10:29

Solution


Given:
In this question, we are given
    • The product of integers a and b is negative
    • The sum of the squares of a and b is greater than 0 but less than 86

To find:
We need to determine
    • The maximum value of ab

Approach and Working:
As the product of ab is negative, one of a and b must be negative and both of them are non-zero.
    • Hence, to maximise the value of ab, we must consider the minimum possible value for the positive integer (i.e. +1) and the maximum possible value of the negative integer (i.e. -1).
    • Therefore, the maximum possible value of ab = (1) x (-1) = -1

Hence, the correct answer is option E.

Answer: E

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Originally posted by EgmatQuantExpert on 20 May 2019, 04:25.
Last edited by EgmatQuantExpert on 20 May 2019, 10:29, edited 1 time in total.
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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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New post 20 May 2019, 08:45
EgmatQuantExpert
please see highlighted part , typo error..

EgmatQuantExpert wrote:

Solution


Given:
In this question, we are given
    • The product of integers a and b is negative
    • The sum of the squares of a and b is greater than 0 but less than 86

To find:
We need to determine
    • The maximum value of ab

Approach and Working:
As the product of ab is negative, one of a and b must be negative and both of them are non-zero.
    • Hence, to maximise the value of ab, we must consider the minimum possible value for the positive integer (i.e. +1) and the maximum possible value of the negative integer (i.e. -1).
    • Therefore, the maximum possible value of ab = (1) x (-1) = -1
Hence, the correct answer is option E.

Answer: B

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Re: If product of integers a and b is negative and sum of their squares is  [#permalink]

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Re: If product of integers a and b is negative and sum of their squares is   [#permalink] 20 May 2019, 10:29
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