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-s

FacebookTweetPin

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

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

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

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

Terms of Service
Privacy Policy
Contact: info@tinkutara.com

FacebookTweetPin