If the difference between two positive numbers is 22 and the

Question

If the difference between two positive numbers is 22 and the difference of their squares is 1100, find the difference between their cubes.

a) 38792
b) 42816
c) 43912
d) 44128
e) 45916

KarishmaB · Accepted Answer

a - b = 22 
 a^2 - b^2 = 1100 = (a - b)(a + b) = 22 * 50 
So (a+b) = 50

(a+b)^2 = a^2 + b^2 + 2ab = 2500 
 (a-b)^2 = a^2 + b^2 - 2ab = 484

Adding them, we get  a^2 + b^2 = 2984/2 = 1492

Subtracting them, we get  4ab = 2016  hence, ab = 504

a^3 - b^3 = (a - b)(a^2 + b^2 + ab) = 22*(1492 + 504) = 22 * 1996 = 22 * (2000 - 4) = 44000 - 88 = 43912

Answer (C)

rajatchopra1994 · Answer

Explanation:
x-y=22... Eqn 1
x^2 - y^2 = 1100
(x+y)(x-y) -= 1100 (Putting value of x-y)
therefore, x+y = 50... Eqn 2

Solving eqn 1 & 2, We get values of x & y. 
x=36, y=14
Now , We need x^3 - y^3 = 43912.
Answer is C

no11one · Answer

this is a very lengthy way to solve this que can anyone suggest a simpler solution

ayerhs · Answer

------------
x-y=22... Eqn 1
x^2 - y^2 = 1100
(x+y)(x-y) -= 1100 (Putting value of x-y)
therefore, x+y = 50... Eqn 2

Solving eqn 1 & 2, We get values of x & y.
x=36, y=14

Now 6 has cyclicity of 1 so last digit is 6 and 4 has cyclicity if 3 so last digit is 4.
Helps to eliminate and we are left with A and C.

Till here its alright.
 Bunuel ,  VeritasKarishma  pls suggest a faster way to approach after this step.

imeanup · Answer

Given,
 (x-y) = 22; (x^2-y^2)=1100 
 (x-y)*(x+y)=1100 
 (x+y) = 50 
Solving the equation,  x=36; y=14

(x-y)^3 = x^3-y^3-3xy(x-y) 
 22^3=x^3-y^3-3*36*14*(22) 
 x^3-y^3=22^3+36*42*22 = 22(22^2+36*42)=22(484+1512)=22*1996=43912 
 Ans C

Hope this helps!

crazyBuny · Answer

We need to find 36^3 - 14^3

From cyclicity we know that unit digits can be subtracted and we can get 2. But we have two options i.e. A & C

At a quick glance you will see that the tends place is different. So you can multiply but only get tens and units place and stop the multiplication. from 36^3 you will get 56 and from 14^3 you will get 44.  Now 56-44 is 12 which is option C

AdiUpp1994 · Answer

I liked Karishma’s approach but this is how I solved it:

A^n - B^n is divisible by A-B for all positive integer values of n. This A^3 - B^3 must be divisible by A-B (22). 22’s prime factors are 2 and 11.

All options are even thus we need to pick the one that is also a multiple of 11. Using divisibility rule of 11, C is the only possible choice.

Thus correct answer: C.

Thoughts  VeritasKarishma  ?

Posted from my mobile device

KarishmaB · Answer

Certainly a great approach!

Since a^3 - b^3 = (a - b)(a^2 + b^2 + ab)
(b+22)^3 - b^3 will be divisible by (b+22 - b) i.e. by 22. 
Fortunately, only option (C) is divisible by 11 but then, GMAT is bound to reward lateral thinking, hence, it may not be all that fortunate either!

Fdambro294 · Answer

Start with

X - Y = 22

The difference between the squares of the 2 Numbers is = 1100

(X + Y) (X - Y) = 1100 ———-> substitute (X - Y) = 22

X + Y = (1100) / (22) = 50

X + Y = 50

+ (X - Y = 22)
______________

2X = 72

X = 36 ———> which means Y = 14

Formula: the difference of cubes formula is given by:

(A)^3 - (B)^3 = (A - B) (A^2 + AB + B^2)

This means the difference between the 2 cubes must be divisible by the difference of the original 2 Numbers (A - B)

In this case: X - Y = 22

To be divisible by 22, the number must be divisible by both 2 and 11

Only one answer choice is divisible by 11 ——> answer (c) 43912

(2 + 9 + 4) - (1 + 3) =

(15) - (4) = 11 = multiple of 11 so the integer is divisible by both 11 and 2 (it is even)

Thus it will be divisible by (X - Y) = 22

C must be the answer.

Posted from my mobile device

Abhishek009 · Answer

(a - b) = 22  &  a^2 - b^2 = 1100

Now,  a^2 - b^2 = (a + b)(a - b)

Further,  1100 = (a + b)*22

So,  (a + b) = 50

We, have  (a + b) = 50  &  (a - b) = 22

Hence,  a = 36  &  b = 14

Now, The difference of their cubes will be

36^3 - 14^3 =  Checking the last digit will work wonder here.

6^3  will have units digit as  6  &  4^3  will have units digit as  4

Thus, Units digit of  36^3 - 14^3 = 2 , among the given options only (C) fits in , hence correct answer must be (C)

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