CBSE Class 11-science Answered

a poitive integer has 1001 digits all of which are 1's . When this number is 1001 the remainder is ?

Asked by byaswanth2005 | 13 Jul, 2020, 09:08: PM

Expert Answer

The given positive integer is 111...1(1001 times)

This number can be rewritten as

$stack stack 111....1 with underbrace below with 1001 space times below equals begin inline style sum from straight r equals 0 to 1000 of end style space 10 to the power of straight r equals fraction numerator open parentheses 10 to the power of 1001 minus 1 close parentheses over denominator 10 minus 1 end fraction space space... space left parenthesis As space 1 comma space 10 comma space 100 comma space 1000 comma space.... space forms space straight a space straight G. straight P. space with space straight a equals 1 space and space common space ratio space equals space 10 right parenthesis equals fraction numerator open parentheses 10 to the power of 1001 minus 1 close parentheses over denominator 9 end fraction Snce space 999 cross times 1000 equals 999 comma 999 comma space theremainder space of space 10 to the power of 6 divided space by space 1001 space is space 1 1001 equals 166 cross times 6 plus 5 So 10 to the power of 1001 equals left parenthesis 10 to the power of 6 right parenthesis to the power of 166 cross times 10 to the power of 5 identical to 10 to the power of 5 left parenthesis mod 1001 right parenthesis 10 to the power of 5 equals 99 cross times 1001 plus 901 Therefore space 10 to the power of 1001 identical to 901 left parenthesis mod 1001 right parenthesis 10 to the power of 1001 minus 1 identical to 900 left parenthesis mod 1001 right parenthesis fraction numerator left parenthesis 10 to the power of 1001 minus 1 right parenthesis over denominator 9 end fraction identical to 100 left parenthesis mod 1001 right parenthesis Hence comma space the space remainder space is space 100.$ (10^1001-1)/9 ≡ 100 (mod 1001)