Find the remainder of the division of 1112131 41516171819 by 99
. (b) Find the remainder 111213141516171819 by 101
For a) I think I just need to sum up two digit such as 135 mod 99 =36
b.) sum up three digit each such as
1971 mod 101 then can i sum 197+1 mod 101?
Is this right?
Answer
N=19+18×100+17×1002+16×1003+15×1004+14×1005+13×1006+12×1007+11×1008
a:100≡1 mod99 ⇒ 100^k≡1\mod 99
⇒ N≡(11+12+13+14+15+16+17+18+19) mod 99 ≡135\ mod 99≡36 \ mod 99
b:100≡ -1\ mod 101 ⇒ 100^{2k}≡[(-1)^{2k}=1]\ mod 101; 100^{2k+1}≡[(-1)^{2k+1}=-1]\ mod 101
⇒ N≡(11-12+13-14+15-16+17-18+19) mod 101≡15\ mod 101
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