# Professor Roman gives unusual math quiz ahead of

His usual Monday Morning 8am class. This is for extra AAA credits.

$$A$$, $$B$$, $$C$$ are distinct digits.

$$AA$$, $$BA$$, $$BBAAA$$, $$CBBBAB$$ are distinct numbers.

Please deduce these with concise reasoning from the given relation:

Rearrangement of the terms in the Equation gives interesting Prime Relationship.

$$\bbox[5px,border:2px solid red]{AA^{BA}-\big(BBAAA\cdot CBBBAB\big) = A}$$

• Thx for the edit..looks better
– Uvc
Jun 23 '19 at 9:26
• You're welcome.
– Ak.
Jun 23 '19 at 9:26
• Comment proving that actual 'digits' are impossible. The term $BBAAA\cdot CBBBAB$ has a maximum order of magnitude of $(4+5)+1=10$ where the $1$ is due to carrying. However, if $B>0$, $AA^{BA}$ will have order of magnitude of at least $11$ since its minimum value is $11^{11}$. We cannot have $A=0$ based on the assumption that $0^0$ is not allowed. Thus $B=0$. This leaves us with the expression $$AA^A-(AAA\cdot C000A0)\equiv(11A)^A-111A\cdot(100000C+10A)=A$$ and we can see that $A^A$ must have the same last digit as $A$, forcing $A=5,6,9$. Checking each value proves their impossibility. Jun 23 '19 at 9:46
• Sure..statement regarding prime relationship gives valuable clue...if terms are rearranged..this is the first number that fails it..
– Uvc
Jun 23 '19 at 9:49

The answer uses Roman numerals, as this is a Prof. Roman puzzle.

A = I
B = X
C = L

This gives the equation:

$$2^{11} - 23*89 = 1$$

Reasoning: after guessing roman numerals were involved, the pattern CBBBAB forces AB to be IX or XC or CM etc. This is because BAB forces A to be a one type digit, but BBB means that B can't be a five type digit because something like VVVIV doesn't exist (it would be XVIV). Taking the simplest case of A=I B=X gives $$2^{11} - 23 * (C+39) = 1$$, which means that C must be 50, or L.

• Prof. Roman congratulates you for your fast thinking and awards you extra credit of AAA points..
– Uvc
Jun 23 '19 at 9:41
• @Uvc do you mean B? how can you award AAA points??? Jun 23 '19 at 9:42
• I mean B..thx..regarding lateral thinking..probably debatable? I wanted to hint that it is not regular number substitution for the letters
– Uvc
Jun 23 '19 at 9:45
• AAA is 3 points?
– JS1
Jun 23 '19 at 9:52
• After solving..it is 111 points...in the puzzle it should mean triple A credits rather than points strictly to avoid confusion
– Uvc
Jun 23 '19 at 10:26