If x is an integer greater than 3, which of the following MUST be divi

Question

If x is an integer greater than 3, which of the following MUST be divisible by 3?

A.  x^3+3x^2−10x

B.  x^3+5x^2+6x

C.  x(x+1)^2

D.  x^3+3x^2−18x

E.  x(x−1)^2

JeffTargetTestPrep · Accepted Answer

We can simplify the expressions by removing multiples of 3 from them. For example, in choice A, the second term 3x^2 is a multiple of 3 regardless what x is. Similarly, -9x is also a multiple of 3 regardless what x is. Therefore, we can remove 3x^2 and -9x from choice A and it leaves us with x^3 - x.

Notice that x^3 - x can be factored as:

x(x^2 - 1)

x(x - 1)(x +1)

(x - 1)(x)(x + 1)

Notice that this is a product of three consecutive integers; thus it’s divisible by 3! = 6. If it’s divisible by 6, then it’s divisible by 3.

Answer: A

pushpitkc · Answer

When we factorize equation,  x^3+3x^2−10x  we get x(x+5)(x-2).

Whatever the value of x(eg 4,5,6), we will definitely get a number divisible by 3.

Hence Option A is the answer.

Abhishek009 · Answer

Thats the perfect way and given this problem during the actual test , I will approach it this way..

However plugging in some numbers can work here...

If x = 4;

4^3+3*4^2−10*4 = 64 + 48 - 40 = 72 , divisible by 3

If x = 5;

5^3+3*5^2−10*5 = 125 + 75 - 20 = 180 , divisible by 3
 
Hence, option (A) works good...

brs1cob · Answer

is there a better solution to this problem.

plug and play is not suitable n it is time-consuming.

felipetakaoka · Answer

KM2018AA   brs1cob

My approach 
Note that x can be either of 3k, 3k+1, or 3k+2, where k integer > 1. This means that each root of the polynomial must yield an integer with a different remainder when divided by 3.
Then do the following
1. Factor the polynomial
2. Check for every option if there is always a factor divisible by 3

Alternative A 
1. x^3+3x2−10x = x(x+5)(x-2)
2. Check
 - If x = 3k, then x(x+5)(x-2) = (3k)(3k+5)(3k-2), which is divisible by 3, since 3k is
 - If x = 3k+1, then x(x+5)(x-2) = (3k+1)(3k+6)(3k-1), which is divisible by 3, since 3k+6 = 3(k+2) is
 - If x = 3k+2, then x(x+5)(x-2) = (3k+2)(3k+7)(3k), which is divisible by 3, since 3k is

Also, note that you can quickly eliminate a few alternatives by noting that the polynomial have repeated roots, such as alternative C and E.

Adarsh_24 · Answer

x(x+5)(x-2)

we know x(x+1)(x+2) is divisible by 3.
x+5 is 3 away from x+2.
x-2 is 3 away from x+1.
so I believe they are equivalent when considering div by 3.

bumpbot · Answer

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If x is an integer greater than 3, which of the following MUST be divi

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If x is an integer greater than 3, which of the following MUST be divi

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