Hi,
(k + 2)(k^2 + 4k + 3) = (k+2)(k+1)(k+3)
From(1): k = 8p
So, (k+1)(k+2)(k+3) = (8p+1)(8p+2)(8p+3) = 2.(8p+1)(4p+1)(8p+3) = 2*odd*odd*odd.
Divisible by 2 but not by 4
Sufficient
From(2): (k + 1)/3 is odd
Let (k+1)/3 = 2p+1 => k = 6p+2.
So, (k+1)(k+2)(k+3) = (6p+3)(6p+4)(6p+5)
if p=1, 6p+4 = 10(not divisible by 4)
if p=2, 6p+4 = 16(divisible by 4). So, we can't say for sure whether it is divisible by 4 or not
Insufficient
Hence, A
Divisibility Difficulties
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Stat.(1): k is divisible by 8 ---> if k is divisible by 8, it's also divisible by 4 (every multiple of 8 is also a multiple of 4).
If k is divisible by 4, then k+2 isn't divisible by 4.
k^2+4k+3 must be odd, since k^2 is even, 4k is even, and 3 is odd; even+even+odd= odd.
Since both (k+2) and (k^2+4k+3) are not divisible by 4 (and are not both even), their product is not divisible by 4. Therefore, the answer is 'no' and the statement is sufficient.
Another option is factoring (k^2+4k+3) into (k+1)(k+3). Thus, (k + 2)(k^2 + 4k + 3) becomes (k+1)(k+2)(k+3): the product of 3 consecutive integers. Since k itself is a multiple of 4, the three following consecutive integers aren't, and only one of them is even. Thus, their product isn't a multiple of 4 (if necessary, try an example such as 9*10*11).
Stat. (2): (k+1)/3 is an odd integer ---> this means that k is an even number, smaller by 1 then a multiple of 3. Thus, k can be a multiple of 4, such as 8 or 20, but also an even number not divisible by 4, such as 2 or 14.
If k is a multiple of 4, the product of (k+1)(k+2)(k+3) is not a multiple of 4 (as explained above), and the answer is 'no'.
If k is an even number not divisible by 4, then k+2 is divisible by 4, and therefore the entire product is divisible by 4. In this scenario, the answer is 'yes'. Hence, the statement is insufficient. This is also derivable by plugging in k=2 and k=8, which lead to 'yes' and 'no' respectively.
The correct answer choice is A.
If k is divisible by 4, then k+2 isn't divisible by 4.
k^2+4k+3 must be odd, since k^2 is even, 4k is even, and 3 is odd; even+even+odd= odd.
Since both (k+2) and (k^2+4k+3) are not divisible by 4 (and are not both even), their product is not divisible by 4. Therefore, the answer is 'no' and the statement is sufficient.
Another option is factoring (k^2+4k+3) into (k+1)(k+3). Thus, (k + 2)(k^2 + 4k + 3) becomes (k+1)(k+2)(k+3): the product of 3 consecutive integers. Since k itself is a multiple of 4, the three following consecutive integers aren't, and only one of them is even. Thus, their product isn't a multiple of 4 (if necessary, try an example such as 9*10*11).
Stat. (2): (k+1)/3 is an odd integer ---> this means that k is an even number, smaller by 1 then a multiple of 3. Thus, k can be a multiple of 4, such as 8 or 20, but also an even number not divisible by 4, such as 2 or 14.
If k is a multiple of 4, the product of (k+1)(k+2)(k+3) is not a multiple of 4 (as explained above), and the answer is 'no'.
If k is an even number not divisible by 4, then k+2 is divisible by 4, and therefore the entire product is divisible by 4. In this scenario, the answer is 'yes'. Hence, the statement is insufficient. This is also derivable by plugging in k=2 and k=8, which lead to 'yes' and 'no' respectively.
The correct answer choice is A.
