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Question Number 21994 by Tinkutara last updated on 08/Oct/17

Suppose N is an n-digit positive  integer such that  (a) all the n-digits are distinct; and  (b) the sum of any three consecutive  digits is divisible by 5.  Prove that n is at most 6. Further,  show that starting with any digit one  can find a six-digit number with these  properties.

$$\mathrm{Suppose}\:{N}\:\mathrm{is}\:\mathrm{an}\:{n}-\mathrm{digit}\:\mathrm{positive} \\ $$$$\mathrm{integer}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{all}\:\mathrm{the}\:{n}-\mathrm{digits}\:\mathrm{are}\:\mathrm{distinct};\:\mathrm{and} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{any}\:\mathrm{three}\:\mathrm{consecutive} \\ $$$$\mathrm{digits}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{5}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:{n}\:\mathrm{is}\:\mathrm{at}\:\mathrm{most}\:\mathrm{6}.\:\mathrm{Further}, \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{starting}\:\mathrm{with}\:\mathrm{any}\:\mathrm{digit}\:\mathrm{one} \\ $$$$\mathrm{can}\:\mathrm{find}\:\mathrm{a}\:\mathrm{six}-\mathrm{digit}\:\mathrm{number}\:\mathrm{with}\:\mathrm{these} \\ $$$$\mathrm{properties}. \\ $$

Commented by Rasheed.Sindhi last updated on 09/Oct/17

Let αabcde     α+a+b=5m.........(i)     a+b+c=5n...........(ii)     b+c+d=5p............(iii)      c+d+e=5q.............(iv)   (i)−(ii):   2−c=5m−5n        c=α−5m+5n        ⋮

$$\mathrm{Let}\:\alpha\mathrm{abcde} \\ $$$$\:\:\:\alpha+\mathrm{a}+\mathrm{b}=\mathrm{5m}.........\left(\mathrm{i}\right) \\ $$$$\:\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{5n}...........\left(\mathrm{ii}\right) \\ $$$$\:\:\:\mathrm{b}+\mathrm{c}+\mathrm{d}=\mathrm{5p}............\left(\mathrm{iii}\right) \\ $$$$\:\:\:\:\mathrm{c}+\mathrm{d}+\mathrm{e}=\mathrm{5q}.............\left(\mathrm{iv}\right) \\ $$$$\:\left(\mathrm{i}\right)−\left(\mathrm{ii}\right):\:\:\:\mathrm{2}−\mathrm{c}=\mathrm{5m}−\mathrm{5n} \\ $$$$\:\:\:\:\:\:\mathrm{c}=\alpha−\mathrm{5m}+\mathrm{5n} \\ $$$$\:\:\:\:\:\:\vdots \\ $$

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