Interesting Observation

[mathnerd_101]

Did you know that any 2-digit number ending with 9 has the property: {AB}=A*B+A+B?
For Example, 69=6x9+6+9!
Can you prove this?

[西瓜]

Proof is below:

AB = 10A + B

10A + B = A*B + A + B

9A = A*B

9 = B

B = 9

[audrey]

@mathnerd_101 nice question! And nice answer @西瓜

Just a tiny note: although your solution is completely correct, for more "formal" proofs you'll want to start from B = 9 and then show how you can manipulate the equation into {AB} = AB + A + B, in order to show that "any 2-digit number ending with 9 (B = 9) has the property {AB} = AB + A + B." That's just because sometimes, when you go from the property you want to prove is true, to the condition (e.g. a 2-digit number ending in 9), you might use a step that isn't reversible, like squaring both sides. (Example: squaring the "equation" 3 = -3 gives 9 = 9, which is true. But clearly, 3 =/= -3!)

In this case, there aren't any irreversible steps - just subtracting something and dividing by something nonzero - it's just interesting to keep in mind.

[西瓜]

@audrey

Thank you very much!

[audrey]

@西瓜 No problem!