Re: For all values of the integer x, is the product (x + 3)(x^2 + 3x + 2)
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03 Aug 2020, 07:51
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 + 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.