$P$ and $Q$ are two positive integers such that $PQ = 64$. Which of the following cannot be the value of $P + Q$?

### Concept & Formula When the product of two variables is given as a constant, their sum can take various values depending on their specific factor pair. We need to find all factor pairs of the given product to determine all possible sums. ### Step-by-Step Solution **Given:** * $P$ and $Q$ are positive integers. * $PQ = 64$ **Calculation / Deduction:** First, identify all pairs of positive integers $(P, Q)$ whose product is $64$. The factor pairs of $64$ are: 1. $1 \times 64 = 64 \implies P+Q = 1+64 = 65$ 2. $2 \times 32 = 64 \implies P+Q = 2+32 = 34$ 3. $4 \times 16 = 64 \implies P+Q = 4+16 = 20$ 4. $8 \times 8 = 64 \implies P+Q = 8+8 = 16$ Comparing these possible sums ($65$, $34$, $20$, $16$) with the given options ($16$, $20$, $35$, $65$), we can see that $35$ is not in our list of possible sums. ### Exam Strategy & Shortcut To quickly check options, mentally deduce the factors of $64$. Notice that $35$ is very close to $34$ (which comes from $32+2$). Since the distance between factors grows rapidly ($32$ to $64$, $16$ to $32$), the resulting sums jump directly from $34$ to $65$. Thus, $35$ is clearly impossible. ### Common Pitfall Students might forget that $P$ and $Q$ can be equal (like $8$ and $8$) unless the question specifically restricts them to "distinct" integers. If they improperly excluded $8+8=16$, they might get confused, but here they can be the same, and $35$ is definitively the impossible value. ### Final Answer Therefore, the correct answer is 35.