f(x)=x^2−1 For which of the following values of x is f(x) divisible by

Question

f(x)=x^2−1. For which of the following values of x is f(x) divisible by 6?

A. 1812
B. 2721
C. 3118
D. 4245
E. 5653

pushpitkc · Accepted Answer

The expression f(x)=x^2−1 = (x+1)(x-1)
Evaluating answer options
A. 1812 : f(x) = (1813)(1811)
B. 2721 : f(x) = (2722)(2720)
C. 3118 : f(x) = (3119)(3117)
D. 4245 : f(x) = (4244)(4246)
E. 5653 : f(x) = (5652)(5654)

Of the answer options, we can reject Options A and C straightaway, 
as they are not divisible by 2.

Careful evaluation of the other answer options : 
Option B = Both 2722 and 2720 are divisible by 2 but not by 3
Option D = Both 4244 and 4246 are divisible by 2 but not by 3.
But, in Option E, 5652 is divisble by 2 and 5654 is divisible by 3.
 Hence Option E is our answer!

Luckisnoexcuse · Answer

Divisible by 6 = Divisible by 2 & Divisible by 3

none of the numbers should be divisible by 3 as they have to follow x^2-1 divisible by 3
only A, C & E left as the sum of their digits is not divisible by 3
B & D are out
now check the last digit of A , C and E
2^2-1 = 3 which is not divisible by 2 ...A out
8^2-1 = 3 which is not divisible by 2 ...C out
3^2-1 = 8 divisible by 2 and the answer

E

septwibowo · Answer

A number is divisible by 6 when the sum of all of the digit is divisible by 3 and the number itself is EVEN.

Try to plugin value X and evaluate its UNIT DIGIT :
A. 1812 ->  (1812^2)-1  --> the result is ODD, not the answer.
B. 2721 -->  (2721^2)-1  --> the result is EVEN, can be the answer.
C. 3118 -->  (3118^2)-1  --> the result is ODD, not the answer.
D. 4245 -->  (4245^2)-1  --> the result is EVEN, can be the answer.
E. 5653 -->  (5653^2)-1  --> the result is EVEN, can be the answer.

As we can see, only B, D and E left. 
At this point, I chose to calculate E :  (4245^2)-1  = 18020024, which sum all of the digits is not divisible by 3. E cannot be the answer.

stuck between B and E.

septwibowo · Answer

Could you please elaborate more the one that I highlighted?

Arsh4MBA · Answer

For a number to be divisible by 6 , check if its divisible by 2,3.

From the given, first filter the numbers that are divisible by 3 as x^2 -1 will definitely be not divible by 3;

This leaves us with A, C , E. Now the square of the last digit and subtract 1. The last digit should be even to be divisble by 6.

A. 1812 -- 2^2-1 =3 not evn
B. 2721 -- Eliminate since divisible by 3.
C. 3118 -- 8^8-1=63 not even
D. 4245 -- Eliminate divisible by 3
E. 5653 -- 8; even

Ans:E

Arsh4MBA · Answer

Nice explaination  pushpitkc

SeanWtx · Answer

If a number that is divisible by 3 is squared, the result is also divisible by 3. So looking at the answer choices, the answer choice cannot itself be divisible by 3 since we will end up subtracting 1 after we square it because after we subtract 1 the result will no longer be divisible by 3.

kshitij89 · Answer

E. 5653 : f(x) = (5654)(5655)

Should be :
E. 5653 : f(x) = (5654)(5652)

5652 is divisible by 2 as well as 3.

Regards
Kshitij

chetan2u · Answer

HI..

as also mentioned above  x^2-1 = (x-1)(x+1) ...
 this should tell us that  ANS will be a number that is NOT div by 3 and 2  
A and C are div by 2 so eliminate
B and D are div by 3, so eliminate
ans E

Reason:-
 1) if x is EVEN, both the number above it and below it will be odd so their product will be ODD and not div by 2
2) If x is div by 3, the number above and below it will not be div by 3

Socalsurfer2020 · Answer

To solve this quickly, we don't need to do any math. We simply have to realize 1) what the numbers are and 2) what the numbers do. For a number to be divisible for 6, it must have at least one factor of two and one factor of three; that is to say it must be even and be divisible by three. It's also important to remember that squaring a number keeps the same prime factors, but simply doubles the values of all of the exponents in the prime factorization.

Rules: For a number to be divisible by 2, it must be even. And for a number to be divisible by 3, the digits in the number must add to equal a number divisible by 3.

With all of this in mind, it's important to realize that any even number minus 1 equals an odd, and that when 1 is subtracted from any multiple of 3, that number is no longer divisible by 3. As such, it follows that when an even multiple of 3 is squared, and then 1 is subtracted, that number is no longer 1) even and 2) divisible by 3, therefore not divisible by 6.

A/C: Both even, therefore, when squared and 1 is subtracted, the outcome will be odd. Eliminate.
B/D: Both are divisible by three, therefore, when they are squared and one is subtracted, they will no longer be divisible by 3. Eliminate.

IMO Answer = E.

minustark · Answer

Since, multiples of 6 ended with even integer, we have to test only the odd options. f(x) =(x+1) (x-1)

B. 2721 = 2722 *2720> not divisible by 3.
D. 4245 = 4244 * 4246 >not divisible by 3.
E. 5653 = 5652*5654 > 5652 is divisible by 6. This is the correct answer.

RajatGMAT777 · Answer

pushpitkc 
Can You correct your explanation 5654 is not divisible by 3

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f(x)=x^2−1 For which of the following values of x is f(x) divisible by

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