When multiplying significant numbers do you round to the least number of significant digits? My textbook says that a problem such as 16.235 × 0.217 × 5 = 17.614975 would be rounded to 20 since twenty has only one significant digit like 5. But this says:

So would a simple problem such as 3 × 1.1 equal 3.3, 3, or 4?

I think your problem belongs more in engineering than in math. The general idea, I would say, is that the accuracy of your result is, at best, as the least accurate measure in your chain. However, what the accuracy of a number actually is, might depend on context. Typically, numbers with decimals carry an expression of their accuracy: 1.200 is more accurate than 1.2, for example. But in the two examples you provide, the integers could mean a different accuracy each. The first example, on your text, might have chosen to write the integer '5' as meaning 'we don't really know if it is 4.9 or 5.3'. Yet in the second example, it is clear from the context than an exact '2' is meant (and not 1.9 or 2.1 hairdryers), especially under the explicit assumption of the hairdryers being identical. So I think both examples might be right, each in its own context, and it is the context which provide the distinction; when in doubt, ask for the origin of the number.

First, the term is "significant digits" not significant numbers.

Your quote is talking about numbers that are derived by counting things, not by measurements. Measurements are necessarily "non-perfect" and have some error. Counting is not: 2 objects are 2 objects without any possibility there being 2.001 or 1.999 objects! The whole point of "significant digits" is that a calculation, based on measurements can't be more accurate than the least accurate measurements. Things based on counting, since they are not measurements don't count toward that.

Unless your textbook is being much too vague, there should be more information than that. In order to apply "significant digits" at all, you need to know where those numbers came from. If this were purely a mathematics problem and those number were "given" then the correct answewr would be 17.614975 and the question of "significant digits" doesn't arise. If all of the numbers, including the 5, are measurements (and writing the 5 like that indicates that the true value might be anywhere from 4.5 to 5.5), then, yes, you would need to write the answer as "20" (better 2.0 x 10) in order not to claim more accuracy than you have. If, however, the "5" is a count, and the other numbers are measurements, You should write the answer as 17.6 since 0.217 has 3 significant figures.

As HallsofIvy points out, there is a difference between counted numbers and measured numbers. An alaming number of textbooks mess up measured numbers and contribute to the difficulty.

If measurements always include the unit (after all, it makes no sense at all to say a rod is 20.34 long), then you can know which factors are exact and which have built-in uncertainity. I even go so far as to retain units like W/W for efficiency (assuming that's how I got the efficiency).

Another problem to watch is numbers with trailing zeros, e.g., 40 m^2 in that you don't know if that is 39.5 - 40.5 m^2 or 35 - 45 m^2. As HallsofIvy again pointed out, scientific notation makes that very clear, i.e., 4.0 X 10 m^2 or 4 x 10 m^2. Another way to make your meaning clearer is to say either 40 m^2 (35-45 m^2) or 40. m^2 (39.5-40.5 m^2) although the trailing decimal point will make some people absolutely bananas.

The number of sig fig depends on context.
If I say I am 3 feet tall, treat that as one sf.
If I say one cake weighs 4.3 pounds, how much do 3 weigh, that 3 has an infinite number of sf. A number like 20 is ambiguous, and also depends on context.
It could have 1, 2 or infinite sf.