Can You Solve the Passcode Riddle?

Here's a nice riddle. Three members of a team have been captured. On their way into the prison (guarded by ravenous mutant salamanders), they pass a series of numbered doorways, each with a keypad featuring the numbers one through nine. Each keypad opens with a code...but you have no idea what that code might be. One member will be allowed to try to escape by facing a challenge. Can the remaining two listen in on that challenge, figure out the correct hallway, and figure out the passcode to open it? With some basic math, they can succeed.

Some more details: Zara, the team member participating in the challenge, has a one-way audio transmitter that allows the other two team members to listen. As Zara is led to the challenge through one of the hallways, she is informed that her challenge is to guess the passcode for her hallway based on rules. Zara is told that the passcode will contain three positive whole numbers, in ascending order (like 1, 2, 3—the second number is greater than or equal to the first, the third likewise to the second).

Zara is told that she may ask up for up to three clues about the code—but she can't say anything else, or else she too will be fed to the mutant salamanders! Through this process of requesting clues and thinking through the problem, Zara implicitly passes information to her compatriots about what the answer is.

Zara asks for the first clue, and is told that the product of the three numbers in the code (x * y * z) is 36. Zara asks for the second clue, and is told that the sum of the numbers in the code (x + y + z) is the same as the number of the hallway she entered. There is a long silence. Then she asks for the third clue, and is told that the largest (greatest) number appears only once in the combination. Shortly after, Zara punches in the code and escapes.

Given that information, can you figure out the passcode? The video below walks through the puzzle and its solution. Here's the text of the puzzle in its simplest form, transcribed from the video (it helpfully asks you to pause before explaining the solution!):

Find three numbers in ascending order! 1. The product of the three numbers is 36. 2. The sum of the three numbers is the same as Zara's hallway number, which you don't know but she does. 3. The largest number must be unique. 4. Zara, a perfect logician, needed clues 1-3 to escape.

Start your figuring!

Then tune in to the video to test your solution:

For a bit more on this puzzle, check out this TED-Ed page, especially the "Dig Deeper" section which contains links to various math resources that you may find useful. (It also includes a link to a puzzle variant that may be interesting after you solve this one.)