2. Nanda and Mike both have a note with the same five-digit number on it. Nanda puts the digit 4 in front of her number and the digit 8 at the end, resulting in a seven-digit number. Mike puts a digit in front of his number. When they compare their new numbers, it turns out that Nanda's number is exactly 6 times larger than Mike's.

What number did they start with?
B2. 49998 We call the five-digit starting number $x$ and the digit Mike adds in front of the number $c$. Thus, Nanda's number is $10 x + 4,000,008$ and Mike's number is $x + c \cdot 100,000$. The fact that Nanda's number is six times as large as Mike's number means that $10 x + 4,000,008 = 6 x + 6 c \cdot 100,000$. Simplifying gives $4 x = c \cdot 600,000 - 4,000,008$ or $x = 150,000 \cdot c - 1,000,002$.

For $c \leqslant 6$ we get $x \leqslant 150,000 \cdot 6 - 1,000,002 = 900,000 - 1,000,002$.

For $c \geqslant 8$ we get $x \geqslant 150,000 \cdot 8 - 1,000,002 = 1,200,000 - 1,000,002 = 199,998$. This is too large since $x$ is a five-digit number. Therefore, it must be that $c < 8$.

The only possibility is thus $c = 7$. For $x$ we then find $x = 150,000 \cdot 7 - 1,000,002 = 49,998$.