Number Theory
# Divisibility

If the number \(\overline{7448x24y}\) is divisible by \(72\), and \(x \ne y,\) find \(x-y\).

\[ {\huge \color{blue}N = \color{green} {123,4}\color{red}{F}\color{green}{6,789}}\]

If \(\color{red} {F}\) is replaced with a digit from the set (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9) at random, what is the probability that \(\color{blue} {N}\) will be divisible by 3?

A longevity chain is a sequence of consecutive positive integers, whose digit sums are never a multiple of 9. What is the longest possible length of a longevity chain?

For similar problems, you can read my note on Construction.

\[ \begin{array} { l l l l } & && & A & A & A \\ \times & & & & A & A & A \\ \hline &A & A & 8 & 0 & 0 & 1\\ \end{array} \]

What is the three digit number \(\overline{AAA}?\)

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