Cracking the Code: Can You Beat the NINJA SENSEI?

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The discussion revolves around a code-cracking challenge involving a PIRATE and NINJA SPIES. The PIRATE inputs a third number based on two given numbers, with examples showing a pattern where the third number seems to relate to the product of the first two numbers plus their sum. The NINJA SENSEI successfully deduces the code and encounters the numbers 3 and 15. Participants suggest using a matrix to identify patterns and consider the factors of previous results like 9, 24, and 45 to find the solution. The conversation emphasizes problem-solving strategies for mathematical puzzles.
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The PIRATE protects the control panel from NINJA SPIES by using a super secret code that only he knows how to work out. When logging into the control panel, he is given two numbers by the servers, but he can't log in until he types in the third number, which is related to the first two.

However, some NINJA SPIES have kept an...er...eye on him as he entered the secret code several times. For example, when 2 and 3 appeared on the screen, he typed in 9. When 5 and 6 appeared, he typed in 24. When 8 and 9 appeared, he typed in 45.

The NINJA SPIES took this information back to their NINJA SENSEI, who cracked the code. The NINJA SENSEI was able to access the login screen, and when she did, the numbers 3 and 15 appeared.

What number did the NINJA SENSAI type to gain access to the control panel?
So my teacher gave me this problem to solve over the weekend for extra credit. I still have yet to solve it. Any clues like equations please? Thanks in advance. :D
 
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Have you tried setting up a matrix of screen numbers and typed numbers to try to discern a pattern? Usually problems like this yield to the scratch-paper approach pretty easily.
 
You might want to think about factors of the result numbers, 9, 24, 45.
 

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