11.8. A student was supposed to multiply a three-digit number by a four-digit number (the numbers do not start with 0). Instead of doing so, he simply appended the four-digit number to the right of the three-digit number, obtaining a seven-digit number, which is $N$ times ($N \in \mathbb{Z}$) larger than the correct result.

a) Can $N$ be equal to 2 or 10?

b) Find the largest possible value of $N$.
Solution. Let $\overline{abc}$ be the three-digit number and $\overline{defg}$ be the four-digit number. Then the seven-digit number has the form $\overline{abcdefg}$. From the condition of the problem, we have $\overline{\overline{abcdefg}}=N \cdot \overline{abc} \cdot \overline{defg}$, which is equivalent to $\overline{abc} \cdot 10^{4}=\overline{defg} \cdot(N \cdot \overline{abc}-1)$ (1).

a) Let $N=2$. Then (1) takes the form $\overline{abc} \cdot 10^{4}=\overline{defg} \cdot(2 \cdot \overline{abc}-1)$ (2). Since $(2 \cdot \overline{abc}-1)$ is an odd number, and $\overline{defg}$ does not divide $10^{4}$, then $(2 \cdot \overline{abc}-1)$ must divide a number of the form $5^{m}(m \geq 1)$. For example, let $2 \cdot \overline{abc}-1=5^{4}=625$. Then $\overline{abc}=313$ and, from (2), we obtain $\overline{defg}=5008$, i.e., $\overline{\overline{abcdefg}}=3135008$. Therefore, the value $N=2$ is possible.

For $N=10$ the equality (1) takes the form $\overline{abc} \cdot 10^{4}=\overline{defg} \cdot(10 \cdot \overline{abc}-1)$. The expression $(10 \cdot \overline{abc}-1)$ is an odd number (does not divide by 2) and has the last digit 9 (does not divide by 5). Therefore, $10^{4}$ must divide the number $\overline{defg}\frac{10000}{\overline{defg}}>\frac{10000}{2000}=5$ and, since $N<10$, it follows that $N \in\{6,7,8,9\}$.

Let $N=9$. Then from (4) we get $\frac{10000}{9}<\overline{defg}<\frac{10020}{9}$, which implies $1111<\overline{defg} \leq 1113$, i.e., $\overline{defg} \in\{1112,1113\}$. Considering $\overline{defg}=1112$, from relation (3) we get $1112=\overline{abc} \cdot 8$, i.e., $\overline{abc}=139$. Therefore, $\overline{abcdefg}=1391112$ and $N_{\max }=9$.