If a and b are integers such that a

Question

If  a  and  b  are integers such that  a < b < 0  and  a^2 = b^2 + 7 , what is the value of  a-b ?

A. -7
B. -4
C. -3
D. -1
E. 1

Bunuel · Accepted Answer

Official Solution:

If  a  and  b  are integers such that  a < b < 0  and  a^2 = b^2 + 7 , what is the value of  a-b ?

A. -7
B. -4
C. -3
D. -1
E. 1

Given:  a^2=b^2+7 .

Rearrange and apply  a^2 - b^2 = (a - b)(a + b)  to get  (a - b)(a + b) = 7 . Since  a  and  b  are integers, both  a + b  and  a - b  are integers as well. Thus, we have the product of two integers equal to 7. There are only two combinations of such factors possible: (1, 7) and (-1, -7). Given that both  a  and  b  are negative, the first case is not possible, so  a - b  must be either -1 or -7. However, it cannot be -7 because in this case,  a + b  must be -1, and no two negative integers yield a sum of -1. Therefore,  a - b = -1 .

Answer: D

gmatophobia · Answer

On a number-line,

------ a -------- b ------- 0 ---------

Inference: Both  a  and  b  are negative.

a^2 = b^2 + 7

a^2 - b^2 = 7

(a+b)(a-b) = 7

The following possibilities hold:

(a+b)  (a-b)  (a+b)*(a-b)  Comments    1  7  7  Invalid as (a+b) should be negative    7  1  7  Invalid as (a+b) should be negative    -1  -7  7  Invalid as (a+b) should be less than -1    -7  -1  7  Valid

Out of the above 4 possibilities, only one of them is valid.

Hence, (a-b) = -1

Option D

GMATinsight · Answer

Note, a and b are integers i.e. a+b as well as a-b are Integers

a^2 = b^2 + 7 
 a^2 - b^2 = 7 
 (a - b)*(a + b) = 7 
i.e. a-b could be {1, 7, -1, -7}

Since  a < b < 0 
- (a+b) must be negative hence (a-b) also must be negative since (a+b)*(a-b) = 7
- Also, a-b must be greater than a+b (since both a and b are negative)

i.e. a-b could be {-1}

Answer: Option D

Nidzo · Answer

a^2=b^2+7

a^2-b^2=7

(a+b)(a-b)=7

1. As we are multiplying the sum of two integers with their difference to get 7, a prime number, we know that one bracket must equal  -7  or  7  and that the other must equal respectively  -1  or  1

2.  We are told that  a<b<0 . Which means that one bracket will equal  -7  and the other  -1 . It also means that,  (a+b)  will be the smaller value as  (a+b)  really is a negative number being subtracted from another negative number, while  (a-b)  will be the larger value as here one is adding a positive number to a negative number.

Therefore,  (a-b) = -1

ANSWER D

VivekSri · Answer

bumpbot · Answer

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If a and b are integers such that a < b < 0 and a^2 = b^2 +

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(a+b)	(a-b)	(a+b)*(a-b)	Comments
1	7	7	Invalid as (a+b) should be negative
7	1	7	Invalid as (a+b) should be negative
-1	-7	7	Invalid as (a+b) should be less than -1
-7	-1	7	Valid

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