MathDB

Super number is a sequence of numbers

0,1,2,\ldots,9

such that it has infinitely many digits at left. For example

\ldots 3030304

is a super number. Note that all of positive integers are super numbers, which have zeros before they're original digits (for example we can represent the number

4

\ldots, 00004

). Like positive integers, we can add up and multiply super numbers. For example:

\begin{array}{cc}& \ \ \ \ldots 3030304 \\ &+ \ldots4571378\\ &\overline{\qquad \qquad \qquad }\\ & \ \ \ \ldots 7601682 \end{array}

And

\begin{array}{cl}& \ \ \ \ldots 3030304 \\ &\times \ldots4571378\\ &\overline{\qquad \qquad \qquad }\\ & \ \ \ \ldots 4242432 \\ & \ \ \ \ldots 212128 \\ & \ \ \ \ldots 90912 \\ & \ \ \ \ldots 0304 \\ & \ \ \ \ldots 128 \\ & \ \ \ \ldots 20 \\ & \ \ \ \ldots 6 \\ &\overline{\qquad \qquad \qquad } \\ & \ \ \ \ldots 5038912 \end{array}

a) Suppose that

A

is a super number. Prove that there exists a super number

B

such that

A+B=\stackrel{\leftarrow}{0}

(Note:

\stackrel{\leftarrow}{0}

means a super number that all of its digits are zero).
b) Find all super numbers

A

for which there exists a super number

B

such that

A \times B=\stackrel{\leftarrow}{0}1

(Note:

\stackrel{\leftarrow}{0}1

means the super number

\ldots 00001

).
c) Is this true that if

A \times B= \stackrel{\leftarrow}{0}

, then

A=\stackrel{\leftarrow}{0}

B=\stackrel{\leftarrow}{0}

? Justify your answer.

Problem Statement