What is the absolute difference between the cubes of two

Question

What is the absolute difference between the cubes of two different non-negative integers?

(1) One of the integers is 2 greater than the other integer.

(2) The square of the sum of the integers is 49 greater than the product of the integers.

Bunuel · Accepted Answer

What is the absolute difference between the cubes of two different non-negative integers?

Given:  x  and  y  are different  non-negative integers . Question:  |x^3-y^3|=?

(1) One of the integers is 2 greater than the other integer -->  |x-y|=2 . There exist infinitely many such  x  and  y , thus we'll have infinitely many different values of  |x^3-y^3| . For example consider  x=2 ,  y=0  and  x=3 ,  y=1 . Not sufficient.

(2) The square of the sum of the integers is 49 greater than the product of the integers -->  (x+y)^2=xy+49  -->  x^2+xy+y^2=49 . Now, we can not simply say that this statement is not sufficient (as done in the above post), because we have some constraints on  x  and  y , namely that they are  non-negative integers . Taking this into account, from  x^2+xy+y^2=49  we can say that both  x  and  y  are integers from 0 to 7, inclusive. By trial and error we can easily find that only two pairs satisfy this equation: (7,0) and (5,3), in any order. Not sufficient.

Notice that if we were told that  x  and  y  are  positive  integers (instead of  non-negative  integers) then this statement would be sufficient, since only one pair would remain: (5,3).

(1)+(2) Since from (1)  |x-y|=2  then from (2) only one pair is valid: (5,3). Sufficient.

Answer: C.

Hope it's clear.

boomtangboy · Answer

Hi,

I got the answer as A.

Let the 2 nos be x & y

We need to find |x^3 - y^3|

Statement 1 : x = 2+y

substitute in quest stem => |(2+y)^3 - y^3| 
from the above you can find y & hence x

Statement 2 : (x+y)^2 - xy > 49
In statement 2 we cant eliminate x or y hence Statement 2 is insufficient

Thus answer is A

Hope this helps

jach2012 · Answer

Statement 2 : (x+y)^2 - xy= 49
on simplification we get  x^2+y^2+xy=49 . Not sufficient.

But when we combine 1&2
 |x^3 - y^3|=|(x-y)(x^2+y^2+xy)| 
 =|(2)(49)|
so we can answer by 1&2 together. So the answer should be C. |

rajeevrks27 · Answer

IMO the correct answer is C
first the requirement is to find the value of |x^3 - y^3| or |(x-y)( x^2 + y^2 + xy)|
1) x = y + 2 or x -y = 2, Not sufficient
2) (x+y)^2 = 49 + xy, as we simplify we get x^2 + y^2 + xy = 49
insufficient because we don't know the value of x - y from this statement
combine 1 and 2, we get the value of (x-y) and ( x^2 + y^2 + xy) hence sufficient.

rajeevrks27 · Answer

Hey Bunnel, thanx for a nice explanation . I have learned something new today with ur explanation. Every time i read ur posts, it further reimposes my faith that ur a true legend.
Thanx a ton.

Smita04 · Answer

Thanks a ton Bunuel. Your explanations are really helpful. Thanks a lot again!

boomtangboy · Answer

Ya thanks Bunuel....

I forgot the expansion of (a^3 - b^3)

galiya · Answer

Could you please help me to understand where is the flaw in my reasoning:
I chose the answer D for this problem, since
the first statement gives us the relationship between two variables : if the second item is two more than the first one, doesn't it mean x=x+2, where x+2 =y

Bunuel · Answer

Merging similar topics. Please refer to the solutions above and ask if anything remains unclear.

galiya · Answer

But, Bunuel, why can't i set x=x+2, where x+2=y? Since the first statement tells us "one of the integers is 2 greater than the other". The only thing confused me - "ONE of the integers", what means i don't know the exact order
May be my question is stupid one- but i just need to grasp it fully

Bunuel · Answer

First of all x=x+2 doesn't make any sense, because x's cancel out and we have that 0=2, which is not true.

Next, "one of the integers is 2 greater than the other integer" CAN be expressed as x+2=y, (assuming that y is larger), but even in this case we cannot get the single numerical value of y^3-x^3, for example consider: y=2, x=0 and y=3, x=1.

Hope it's clear.

vdbhamare · Answer

IMO C..

nice work bunnel.....

But are we expected to know 5,3 is the pair which solves given condition if integers are only positive?

Bunuel · Answer

You can get that the statements taken together are sufficient even if you don't know that the only pair possible is (5, 3):  x^3-y^3=(x-y) (x^2+x y+y^2)  --> (1) says that  x-y=2  and (2) says that  x^2+x y+y^2=49  -->  x^3-y^3=(x-y) (x^2+x y+y^2)=2*49=98 .

Hope it's clear.

qlx · Answer

It was difficult to see that  x^3-y^3  can we written as  (x-y) (x^2+x y+y^2)

is there an easier way to understand this or must it be learnt by heart.

Bunuel · Answer

It's good to memorize that formula. But even if you don't know it, you can still the correct answer as shown here: what-is-the-absolute-difference-between-the-cubes-of-two-129157.html#p1059501

himanshujovi · Answer

Hello Bunuel . After solving stem 2 and coming to the point of  x^2+xy+y^2=49  , I deduced that since it is an equation in 2 variables it will not be solvable. I substituted from 1 ( y=x+2) and then realized that it transforms to a simple quadratic and then with the mentioned restriction , came to C. Would this be a right approach ?

Bunuel · Answer

Have you read this:
Now, we can not simply say that this statement is not sufficient (as done in the above post), because we have some constraints on  x  and  y , namely that they are  non-negative integers . Taking this into account, from  x^2+xy+y^2=49  we can say that both  x  and  y  are integers from 0 to 7, inclusive. By trial and error we can easily find that only two pairs satisfy this equation: (7,0) and (5,3), in any order. Not sufficient.

Notice that if we were told that  x  and  y  are  positive  integers (instead of  non-negative  integers) then this statement would be sufficient, since only one pair would remain: (5,3).

masoomjnegi · Answer

Statement 1. One of the integers is 2 greater than the other integer.
x = y + 2. Using this, we cannot find x3 – y3 . Hence, Insufficient.
Statement 2. The square of the sum of the integers is 49 greater than the product of the integers.
(x+y)2 = 49 + xy
x2 + y2 + xy = 49. Using this we can’t find x3 – y3. Hence, Insufficient.
Statement 1 & 2 together. 
x2 + y2 + xy = (y+2)2 + y2 +(y+2)y = 49
y2 + 2y – 15 = 0
This gives y = -5,3. Since y is non negative, we take y = 3 and x = y + 2 = 5. Hence, sufficient.
Hence, c is the answer.

pratiksha1998 · Answer

Hi, For Statements like Statement 2 how to find the pairs quickly. I thought the only pair possible for statement 2 is 0 and 7 and hence choose B. Does someone know a quicker way for finding such pairs?

What is the absolute difference between the cubes of two

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