For all values of the integer x, is the product (x + 3)(x^2 + 3x + 2)

Question

For all values of the integer x, is the product (x + 3)(x^2 + 3x + 2) divisible by 4?

(1) (x + 3)/5 is an odd integer
(2) x is divisible by 16

ProfChaos · Answer

This is a Yes/No question. For sufficiency, a definite "yes" would show that the product (x + 3)(x2 + 3x + 2) is divisible by 4, or a definite "no" would show that it is not.

Let’s simplify the product by factoring it:

(x + 3)(x^2 + 3x + 2)

(x + 3)(x + 2)(x + 1)

If the value of x makes any of these three factors divisible by 4, then the entire product will be divisible by 4.

In any set of four consecutive integers, exactly one will be a multiple of 4. Therefore, if x is a multiple of 4, then neither (x + 3), (x + 2), nor (x + 1) will be. However, if x is not a multiple of 4, exactly one of (x + 3), (x + 2), and (x + 1) must be.

Therefore, for sufficiency we need to know whether x is a multiple of 4.

Evaluate the Statements:

Statement (1): We are told that (x+3)/5 is an odd integer. Let’s translate this into an equation so we can learn about x:

(x+3)/5 = odd

x + 3 = 5(odd)

x = 5(odd) – 3

This tells us that x is even, so it might be a multiple of 4 but also might not. Statement (1) is Insufficient.

Picking Numbers can illustrate this. If = 1, then x + 3 = 5. This yields x = 2. Hence, (x + 3)(x + 2)(x + 1) = (5)(4)(3). This must be divisible by 4, as 4 is one of the terms.

But it could be that = 3, then x + 3 = 15. This yields x = 12. Hence, (x + 3)(x + 2)(x + 1) = (15)(14)(13). This is not divisible by 4.

Since (x + 3)(x + 2)(x + 1) might or might not be divisible by 4, Statement (1) is Insufficient to answer the question with a definite "yes" or a definite "no." Eliminate choices (A) and (D).

Statement (2): We are told that that x is divisible by 16. If x is divisible by 16, we also know that x is divisible by 4, since 16 is a multiple of 4. This is what we needed; Statement (2) is Sufficient.

We can also use Picking Numbers. If x = 16, then (x + 3)(x + 2)(x + 1) = (19)(18)(17). We wouldn’t want to actually calculate this, but there is a way around the calculations. Since 4 = 2 x 2, a number must have at least two factors of 2 to be divisible by 4. The product (19)(18)(17) has only one factor of 2, so it is not divisible by 4.

If x = 32, then (x + 3)(x + 2)(x + 1) = (35)(34)(33), which also has only one factor of 2 and is thus not divisible by 2.

Since (x + 3)(x + 2)(x + 1) cannot be divisible by 4, Statement (2) is Sufficient to answer the question with a definite "no." Eliminate choices (C) and (E).

Therefore, Choice (B) is correct.

MunkhbatBat · Answer

Hi.

I got a question regarding statement 2. x is divisible by 16 means 0,16,32,48 are the potential values for x. But 0 won't make (x+3)*(x+2)*(x+1) divisible by 4.
Hence, stmt 2 is insuffienct, is it not?

Mbat

hemantbafna · Answer

(x+3)(x^2+3x+2)=(x+3) (x+2) (x+1)

We need to check if (x+3) (x+2) (x+1)/4

(1) x+3/5 does not help much with the equation.
Not sufficient.

(2) x is divisible by 16, then x must be some multiple of 16. When we take x as 16, x+3=19, x+2=18, x+1=17
None of the numbers are divisible by 4.
Statement 2 helps us in answering teh question.

Sufficient.

B

drwharton3900 · Answer

you see  MunkhbatBat  16 is even since X is div 4 16 is a multiple of 4 so Stmt2 is sufficient to deduce it will be divisible

MunkhbatBat · Answer

What about 0 as a solution for statement two? 0 is divisible by 16. Yet by substituting 0 into (x+3)*(x+2)*(x+1) we wont get a multiple of 4. Hence stmt two is insufficient?

Can someone help? Am i missing something here?

Thanks

bumpbot · Answer

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nitishjain0109 · Answer

For all values of the integer x, is the product (x + 3)(x2 + 3x + 2) divisible by 4?

nitishjain0109 · Answer

Analyze the Question Stem:

This is a Yes/No question. For sufficiency, a definite "yes" would show that the product (x + 3)(x2 + 3x + 2) is divisible by 4, or a definite "no" would show that it is not.

Let’s simplify the product by factoring it:

(x + 3)(x2 + 3x + 2)

(x + 3)(x + 2)(x + 1)

If the value of x makes any of these three factors divisible by 4, then the entire product will be divisible by 4.

In any set of four consecutive integers, exactly one will be a multiple of 4. Therefore, if x is a multiple of 4, then neither (x + 3), (x + 2), nor (x + 1) will be. However, if x is not a multiple of 4, exactly one of (x + 3), (x + 2), and (x + 1) must be.

Therefore, for sufficiency we need to know whether x is a multiple of 4.

Evaluate the Statements:

Statement (1):  We are told that is an odd integer. Let’s translate this into an equation so we can learn about x:

= odd

x + 3 = 5(odd)

x = 5(odd) – 3

This tells us that x is even, so it might be a multiple of 4 but also might not. Statement (1) is  Insufficient .

Picking Numbers can illustrate this. If = 1, then x + 3 = 5. This yields x = 2. Hence, (x + 3)(x + 2)(x + 1) = (5)(4)(3). This must be divisible by 4, as 4 is one of the terms.

But it could be that = 3, then x + 3 = 15. This yields x = 12. Hence, (x + 3)(x + 2)(x + 1) = (15)(14)(13). This is not divisible by 4.

Since (x + 3)(x + 2)(x + 1) might or might not be divisible by 4, Statement (1) is Insufficient to answer the question with a definite "yes" or a definite "no." Eliminate choices (A) and (D).

Statement (2):  We are told that that x is divisible by 16. If x is divisible by 16, we also know that x is divisible by 4, since 16 is a multiple of 4. This is what we needed; Statement (2) is  Sufficient .

We can also use Picking Numbers. If x = 16, then (x + 3)(x + 2)(x + 1) = (19)(18)(17). We wouldn’t want to actually calculate this, but there is a way around the calculations. Since 4 = 2 x 2, a number must have at least two factors of 2 to be divisible by 4. The product (19)(18)(17) has only one factor of 2, so it is not divisible by 4.

If x = 32, then (x + 3)(x + 2)(x + 1) = (35)(34)(33), which also has only one factor of 2 and is thus not divisible by 2.

Since (x + 3)(x + 2)(x + 1) cannot be divisible by 4, Statement (2) is Sufficient to answer the question with a definite "no." Eliminate choices (C) and (E).

Therefore, Choice (B) is correct.

For all values of the integer x, is the product (x + 3)(x^2 + 3x + 2)

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