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## Divisibility

You can divide 6 into equal parts of 1, 2, 3, or 6 (but not 4 or 5) because 6 is divisible by these numbers. The rules of divisibility have wide-ranging applications as an easy test for divisibility.

# Level 3

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.

As $$n$$ ranges over all real values in the interval $$[0, 100]$$, how many values of $$n$$ are there such that $$4n+ 1$$ is an odd integer?

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

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

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