Number Theory

Divisibility

Divisibility: Level 3 Challenges

         

If the number 7448x24y\overline{7448x24y} is divisible by 7272, and xy,x \ne y, find xyx-y.

N=123,4F6,789 {\huge \color{#3D99F6}N = \color{#20A900} {123,4}\color{#D61F06}{F}\color{#20A900}{6,789}}

If F\color{#D61F06} {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 N\color{#3D99F6} {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 nn ranges over all real values in the interval [0,100] [0, 100] , how many values of nn are there such that 4n+1 4n+ 1 is an odd integer?

AAA×AAAAA8001 \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 AAA?\overline{AAA}?

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