If P is divisible by 2, is Q + 5 an integer? (1) The median

If P is divisible by 2, is Q + 5 an integer?

P is divisible by 2 implies that P is an even integer --> P=2*integer, for some integer k.

(1) The median of P and Q is not an integer. The median of P and Q is \(\frac{P+Q}{2}=\frac{2*integer+Q}{2}=integer+\frac{Q}{2}\). So, we are given that \(integer+\frac{Q}{2}\) is not an integer. This means that Q/2 is not an integer, thus Q is not an even integer, which means that Q is either an odd integer or not an integer at all. Not sufficient.

(2) The average (arithmetic mean) of 3P, Q, and Q + 10 is an even integer --> \(\frac{3P+Q+Q + 10}{3}=even\) --> \(3P+2Q+10=3*even=even\) --> \(2Q=even-3P-10=even-even-even=even\) --> \(2Q=even\) --> \(Q=\frac{even}{2}=integer\). Sufficient.

Answer: B.
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2)
(3P+Q+Q+10)/3=3P/3+2(Q+5)/3
3P/3=P(Even integer)
2(Q+5)/3=Even Integer {:Note: P=Even; Even+Even=Even}
2(Q+5)=3*Even integer
Q+5=3*Even Integer/2=3*Integer{:Note: All Even integers contain at least ONE 2}
"Q+5" must be an integer. {:Note: 3*Integer=Integer; Integer * Integer= Integer}
Sufficient.

Ans: "B"

Yes, P divisible by 2 means P is even integer.
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Date - P is even. The question is basically - Does Q is an integer?

So - Statement 1 - INSU.
Lets say P=2,Q=3 ----> The answer is YES.
P=2, Q=2.5 ----> The answer is NO.

statement 2 - SUFF.
\((3p+q+q+10)/3 = Integer\)

3p/3 = p ----> Integer.

2Q+10/3 = Integer. we know that 10/3 is not an integer. therefore 2Q is not an integer as well.
so 2q=not integer.
Q = not integer/2 ----> Q is not an integer.

could there be more discussion on this one? I don't get how you go from knowing 10/3 is not an integer to knowing that 2Q is not an integer either...

Hi
Can you find what's wrong with my reasoning?

(3Q/3) + (2Q+10)/3 = even int
Q + (Q+5)/3 = even int
Q+5 = even int -Q = even int
Q = even int- 5 = odd int.

How did you get (Q+5)/3 from (2Q+10)/3 ?
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Sorry... my mistake

P is divisible by 2 implies that P is an even integer --> P=2*integer, for some integer k.

If P= 4.4 then still P is divisible by 2.
Now, where am I wrong?

chetan2u Bunuel - When we say "P is divisible by 2", can we safely assume that P is an integer and not a decimal number because even 2.2 is divisible by 2.

Thank you.

For option 2-

If \(\frac{(3P + 2Q + 10)}{2}\) is even, then 3P + Q + 5 must be even (once the 2 in the denominator has canceled out with the 2 in each part of the trinomial in the numerator. Moreover, they have to add up to an integer so if P is an integer so so must Q be.