Question

If $a^{b^c}=256$, then find the maximum possible value of $abc$, where $a, b$ and $c$ are positive integers.

• A. 12
• B. 16
• C. 32
• D. 256

Answer 1

Answer: (C)

32

Answer 2

We are given $a^{b^c}=256$.

Since $256=2^8$, let $a=2^k$ where $k$ is a positive integer.

Then $(2^k)^{b^c}=2^{k\cdot b^c}=2^8$.

Thus, $k\cdot b^c=8$.

The factor pairs for 8 are:

• $k=1$ and $b^c=8$
• $k=2$ and $b^c=4$
• $k=4$ and $b^c=2$
• $k=8$ and $b^c=1$.

Now, analyze each case for the value of $abc$. Note that $a=2^k$:

1. For $k=1$: $a=2$ and $b^c=8$.

   Possible pairs for $(b,c)$:

   • $b=8, c=1$ giving $abc=2\cdot8\cdot1=16$.
   • $b=2, c=3$ giving $abc=2\cdot2\cdot3=12$.

   Maximum here: 16.

2. For $k=2$: $a=4$ and $b^c=4$.

   Possible pairs:

   • $b=4, c=1$ giving $abc=4\cdot4\cdot1=16$.
   • $b=2, c=2$ giving $abc=4\cdot2\cdot2=16$.

   Maximum here: 16.

3. For $k=4$: $a=16$ and $b^c=2$.

   Since $b^c=2$ only possibility is $b=2, c=1$ yielding $abc=16\cdot2\cdot1=32$.

4. For $k=8$: $a=256$ and $b^c=1$.

   The only possibility is $b=1$. Although $1^c=1$ for any $c$, selecting larger $c$ would make $abc$ arbitrarily large. However, given the multiple choice options (12, 16, 32, 256), we must consider the non‐trivial cases ($b>1$).

Thus, among the non‐trivial solutions, the maximum possible value of $abc$ is $32$.