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can-you-find-any-arrangement-of-nine-digits-of-1-9-such-as-967854312-and-the-first-digit-should-be-divisible-by-1-thefirst-two-digitds-should-be-divisible-by-2-the-first-three-digitds-should-be-divisi




Question Number 208555 by liuxinnan last updated on 18/Jun/24
can you find any arrangement of nine digits of 1−9  such as 967854312  and the first digit should be divisible by 1  thefirst two digitds should be divisible by 2  the first three digitds should be divisible by 3   ......  the number should be fivisible by 9  if is such a number available
$${can}\:{you}\:{find}\:{any}\:{arrangement}\:{of}\:{nine}\:{digits}\:{of}\:\mathrm{1}−\mathrm{9} \\ $$$${such}\:{as}\:\mathrm{967854312} \\ $$$${and}\:{the}\:{first}\:{digit}\:{should}\:{be}\:{divisible}\:{by}\:\mathrm{1} \\ $$$${thefirst}\:{two}\:{digitds}\:{should}\:{be}\:{divisible}\:{by}\:\mathrm{2} \\ $$$${the}\:{first}\:{three}\:{digitds}\:{should}\:{be}\:{divisible}\:{by}\:\mathrm{3}\: \\ $$$$…… \\ $$$${the}\:{number}\:{should}\:{be}\:{fivisible}\:{by}\:\mathrm{9} \\ $$$${if}\:{is}\:{such}\:{a}\:{number}\:{available} \\ $$
Answered by nikif99 last updated on 18/Jun/24
5th digit is 5.  2nd, 4th, 6th, 8th digits are even.  1st, 3rd, 7th, 9th digits are odd.  4th is 2 or 6 (ending in 12, 16, 32, 36...)  (1st+2nd+3rd) as well (4th+5th+6th)  must be divided by 3.  ...  By eliminating some cases and “try  and check”, you reach 381654729.  PS. Then, you add a 0 and you obtain  a number duvisible by 10.
$$\mathrm{5}{th}\:{digit}\:{is}\:\mathrm{5}. \\ $$$$\mathrm{2}{nd},\:\mathrm{4}{th},\:\mathrm{6}{th},\:\mathrm{8}{th}\:{digits}\:{are}\:{even}. \\ $$$$\mathrm{1}{st},\:\mathrm{3}{rd},\:\mathrm{7}{th},\:\mathrm{9}{th}\:{digits}\:{are}\:{odd}. \\ $$$$\mathrm{4}{th}\:{is}\:\mathrm{2}\:{or}\:\mathrm{6}\:\left({ending}\:{in}\:\mathrm{12},\:\mathrm{16},\:\mathrm{32},\:\mathrm{36}…\right) \\ $$$$\left(\mathrm{1}{st}+\mathrm{2}{nd}+\mathrm{3}{rd}\right)\:{as}\:{well}\:\left(\mathrm{4}{th}+\mathrm{5}{th}+\mathrm{6}{th}\right) \\ $$$${must}\:{be}\:{divided}\:{by}\:\mathrm{3}. \\ $$$$… \\ $$$${By}\:{eliminating}\:{some}\:{cases}\:{and}\:“{try} \\ $$$${and}\:{check}'',\:{you}\:{reach}\:\mathrm{381654729}. \\ $$$${PS}.\:{Then},\:{you}\:{add}\:{a}\:\mathrm{0}\:{and}\:{you}\:{obtain} \\ $$$${a}\:{number}\:{duvisible}\:{by}\:\mathrm{10}. \\ $$

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