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 A097077 a(n) has the property that when multiplied by an appropriate n-digit number the product is the n-digit number repeated 6 times. 0
 111111, 10101010101, 1001001001001001, 100010001000100010001, 10000100001000010000100001, 100000010000001000000100000010000001, 10000000100000001000000010000000100000001 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS These numbers are divisible by 7 if the number of repeat digits d, is not a multiple of 6. Given the Sum = ((10^d)^6 - 1))/(10^d-1) from Fermat's little theorem, we note that (10^d)^(7-1) == 1 (mod 7) as required. The case when the repeat digits d = 6k we have the denominator of the Sum (10^k)^6 - 1 which by Fermat's little theorem again is divisible by 7. Thus 7 is canceled from the Sum and the sum is then not divisible by 7. It remains to prove this last statement. Also it appears 7 divides (10^k)^6 q+1 times when k = a*7^q. Obviously this method could be extended to divisibility by any prime. REFERENCES The question arose on the mathforfun(AT)yahoogroups.com. Mansoor Anees asked: When an integer or a set of integers is repeated six times to form a new integer, it is divisible by 7. Why? For example, 121212121212 and 123123123123123123 and 444444 are divisible by 7. LINKS FORMULA Sum = (10^d)^5 + (10^d)^4 + . . . + (10^d)^1 + 1 = ((10^d)^6 - 1))/(10^d-1). EXAMPLE If n = 2 we have the number 10101010101. Multiply this by 13 to get 131313131313. and 131313131313/7 = 18759018759. PROG (PARI) divseven(n) = { for(x=1, n, y=((10^x)^6-1)/(10^x-1); if(y%7==0, print1(y", "))) } CROSSREFS Sequence in context: A135403 A154549 A038448 * A109716 A292471 A228253 Adjacent sequences:  A097074 A097075 A097076 * A097078 A097079 A097080 KEYWORD easy,nonn,base AUTHOR Cino Hilliard, Sep 13 2004 STATUS approved

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Last modified July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)