Solution

We have:

\(A=4(10^n)+1,\)

\(B=4(10^{n+2})+1\)

Quick trick: pick \(n=1\)

\(A:\)
\(4(10)+1=41\)

\(41\div7 \text{ leaves } {6}\)

\(B:\)
\(4(10^3)+1\)

Since \((10^3=1000),\) and \((1000\div7)\) leaves (6):

\(4(6)+1=25\)

Remainder \((=4).\)

So for \((n=1): A>B\)

But we need to see whether another positive integer gives the opposite.

Try \((n=2)\)

A:
\(4(100)+1=401\)

Remainder when divided by 7:

\({2}\)

For B, \((n+2=4):\)

\(4(10^4)+1\)

Since \((10^4)\) leaves (4) when divided by 7:

\(4(4)+1=17\)

Remainder =3

Thus: \(A<B\)

We have found both possibilities:

\(n=1\Rightarrow A>B\)
\(n=2\Rightarrow A<B\)

Therefore:

Answer: D