While multiplying four real numbers, John took one of the numbers as 6

Question

While multiplying four real numbers, John took one of the numbers as 62 instead of 26. As a result, the product went up by 720. Then the minimum possible value of the sum of cubes of the other three numbers is

A. 40
B. 50
C. 60
D. 70
E. 80

A. 40
B. 50
C. 60
D. 70
E. 80

LeoN88 · Answer

IMO A

62(abc) - 26(abc) =720
abc(36) = 720

abc= 20

Concept: For abc=k, Sum a+b+c is minimum for a=b=c.

So, required 3 * (20)^(1/3)

How to find the cube root of 20,
cube root of 1=1
Cube root of 8=2
Cube root of 27=3

Let cube root of 20 be 2.5 (Assume) <I took a number on the higher side>

Answer= 3 * (2.5)^(3)~ 3*15 ~45, closest option A.

*Edited* Answer is C.

abc=20
so, a=b=c= (20)^(1/3)
a^3 + b^3 +c^3= 20+20+20= 60

CareerGeek · Answer

Let the product of 3 other numbers = x
So, actual product = 26x
Product taken = 62x 
Difference between the products = 720
—> 62x - 26x = 720
—> 36x = 720
—> x = 20

Possible values = 1*1*20, 2*2*5 ....
Possible sum of cubes = 1 + 1 + 8000 = 8002, 2^3 + 2^3 + 5^3 = 141 . . .

For the sum of cubes of each product to be minimum. All the 3 values must be same

Let each value = a
So, x = a^3 = 20.

Sum of cubes = 3*a^3 = 60.

IMO Option C.

Posted from my mobile device

vishalthomas · Answer

Based on the question, we can form the equation 26XYZ+720= 62XYZ , considering X,Y,Z are the the other 3 numbers.

Therefore, the equation reduces to 36XYZ=720 or XYZ=20.

To get the minimum sum of the cubes we assume we assume that X=Y=Z , and in turn, this means X=Y=Z= cube root (20).

Which implies, the minimum sum of the cubes of X,Y & Z is 60.

hriday17 · Answer

You can take four nos. as a,b,c and 26 and their product will be x
 abc26=x 
we can rewrite it as  abc=x/26 -----------(1)

If digits of 26 are reversed then their product increases by 720-
so we can write that by-  abc62=x+720 
Now lets put in the value of abc from equation (1) here

x/26 * 62=x+720 
x=520
so, abc=20

Now we have to select the real nos., the number will be an integer as the sum of cube of those nos. will be an integer.
There are 2 cases-(not taking + & - ones)
 5*2*2 and 5*4*1 
To make the answer choice possible 2 digits have to be negative (as if we take all the digits as positive the answer will not match)

Case 1) 5*2*2 
Let's take (-5)^3+2^3+(-2)^3=-125, not matching
5^3+(-2)^3+(-2)^3=+125, not matching
so this case is not possible.

Case2) 5*4*1 
(-5)^3+4^3+(-1)^3=-62, not matching
 5^3+(-4)^3+(-1)^3= 60 , This answer is matching the answer choice C
So this is the correct answer.

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While multiplying four real numbers, John took one of the numbers as 6

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