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# If a<b<0 are integers, what is the value of a^3-b^3?

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GMATH Teacher
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Joined: 12 Oct 2010
Posts: 935
If a<b<0 are integers, what is the value of a^3-b^3?  [#permalink]

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03 Mar 2019, 17:21
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GMATH practice exercise (Quant Class 16)

If $$a<b<0$$ are integers, what is the value of $$a^3-b^3$$ ?

(1) $$a^2-b^2 = 65$$
(2) $$a^2 +b^2 < 100$$

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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Re: If a<b<0 are integers, what is the value of a^3-b^3?  [#permalink]

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03 Mar 2019, 19:41
If $$a<b<0$$ are integers, what is the value of $$a^3-b^3$$ ?

So we have to find the value of a and B...

(1) $$a^2-b^2 = 65$$
(a-b)(a+b)=65=(-1)(-65)...so a=-33 and b=-32
But (a-b)(a+b) can be (-5)(-13), so a=-9 and b=-4
Different possibilities..
Insufficient

(2) $$a^2 +b^2 < 100$$
a and B can take various values...
a=-9, b=-1 or a=-8, b=-2, or a=-8, b=-3 and so on
Insufficient

Combined..
$$(-33)^2+(-32)^2>100$$, so eliminate
$$(-9)^2+(-4)^2=81+16=97<100$$, so possible..
Thus $$a^3-b^3=(-9)^3-(-4)^3$$
Sufficient

C
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Re: If a<b<0 are integers, what is the value of a^3-b^3?  [#permalink]

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04 Mar 2019, 00:14
1
fskilnik wrote:
GMATH practice exercise (Quant Class 16)

If $$a<b<0$$ are integers, what is the value of $$a^3-b^3$$ ?

(1) $$a^2-b^2 = 65$$
(2) $$a^2 +b^2 < 100$$

#1
$$a^2-b^2 = 65$$
(a+b)*(a-b)=65
a =-33,b=-32
a=-9, b=-4
in sufficeint
#2
$$a^2 +b^2 < 100$$
we can get many values of a & b to satisfy this relation , viz , 1,2 ; 3,4; 5,4... so on
in sufficient
from 1 & 2
only a= -9 and b= -4 suffices both eqns 1&2
IMO C
GMATH Teacher
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Posts: 935
Re: If a<b<0 are integers, what is the value of a^3-b^3?  [#permalink]

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04 Mar 2019, 05:21
fskilnik wrote:
GMATH practice exercise (Quant Class 16)

If $$a<b<0$$ are integers, what is the value of $$a^3-b^3$$ ?

(1) $$a^2-b^2 = 65$$
(2) $$a^2 +b^2 < 100$$

$$? = {a^3} - {b^3}$$

$$\,a < b < 0\,\,\,{\rm{ints}}\,\,\,\,\left( * \right)$$

$$\left( 1 \right)\,\,\,\left\{ \matrix{ \,\left( {a + b} \right)\left( {a - b} \right) = 65 = 1 \cdot 65 = 5 \cdot 13 \hfill \cr \,\left( * \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,a + b < a - b\,\,\, < 0\,\,\,{\rm{ints}} \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\rm{I}}.\,\,\left\{ \matrix{ \,a + b = - 65 \hfill \cr \,a - b = - 1 \hfill \cr} \right.\,\,\,\,\,\,\,\,{\rm{or}}\,\,\,\,\,\,{\rm{II}}.\,\,\left\{ \matrix{ \,a + b = - 13 \hfill \cr \,a - b = - 5 \hfill \cr} \right.$$

$${\rm{I}}.\,\,\left\{ \matrix{ \,a + b = - 65 \hfill \cr \,a - b = - 1 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\, \ldots \,\,\,\,\, \Rightarrow \,\,\,\,\left( {a,b} \right) = \left( { - 33, - 32} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {\left( { - 33} \right)^3} - {\left( { - 32} \right)^3}$$

$${\rm{II}}.\,\,\left\{ \matrix{ \,a + b = - 13 \hfill \cr \,a - b = - 5 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\, \ldots \,\,\,\,\, \Rightarrow \,\,\,\,\left( {a,b} \right) = \left( { - 9, - 4} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {\left( { - 9} \right)^3} - {\left( { - 4} \right)^3} \ne {\left( { - 33} \right)^3} - {\left( { - 32} \right)^3}$$

$$\left( 2 \right)\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {a,b} \right) = \left( { - 2, - 1} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\left( { - 2} \right)^{\rm{3}}} - {\left( { - 1} \right)^3}\,\, \hfill \cr \,{\rm{Take}}\,\,\left( {a,b} \right) = \left( { - 3, - 1} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\, = {\left( { - 3} \right)^{\rm{3}}} - {\left( { - 1} \right)^3}\,\, \ne \,{\left( { - 2} \right)^{\rm{3}}} - {\left( { - 1} \right)^3}\,\, \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{II}}.\,\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Re: If a<b<0 are integers, what is the value of a^3-b^3?   [#permalink] 04 Mar 2019, 05:21
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