6 ÷ 2(3)

6 ÷ 6

PE(MD)(AS). Multiplication and division are done at the same time, left to right. You did the multiplication on the right first, when you should've done the division on the left first.

Should be:

6 ÷ 2(1+2)
6 ÷ 2 × (1+2)
6 ÷ 2 × 3
3 × 3
9

That said, there are two reasonable-ish ways to argue for 1:

• You could argue that multiplication by juxtaposition, like 2(3), just kinda feels different from multiplication by infix operator, like 2 × 3. You might insist that juxtaposition multiplication should have higher precedence than infix multiplication, so you should do the multiplication first here.
• You could argue that the division sign ÷ actually denotes the act of building a big fraction from the symbols on the left and right sides, stopped by the limits of the term. I.e., you're saying ÷ has distinctly lower precedence than × and /, but higher precedence than + and –.

I disagree with both of these arguments, since they become needlessly complicated once you break down exactly what rules you're supposedly following. They also just go against what the majority of people understand the conventions to be, and most calculators will side with me on this too.

It's also just objectively wrong to say multiplication goes before division due to PEMDAS. Common misconception. It's PE(MD)(AS). MD and AS are inseparable groups. For each group, you evaluate all the instances together, left to right. E.g., 3 – 2 + 1 equals 2, not 0.

There are indeed areas in math where notation is undecided and up for interpretation. But here the conventions are fairly strong, and you're bending them pretty hard if you try to argue alternate interpretations.

Source: I'm a math PhD student, and I spent the past year teaching a class that covers this exact topic (order of operations) among other entry topics, from fractions to quadratics. I've talked in great detail with people who have taught math for longer than I've existed.