Remainder division using total digit

Question

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?

Duplicate of Tests for divisibility by $,9,11,99,101,999,1001$ — Bill Dubuque, Jun 12 '19 at 19:49
See also this answer and search on "casting nines" for many related threads. — Bill Dubuque, Jun 12 '19 at 19:53

score 2 · Answer 1 · answered Jun 12 '19 at 19:18

2

You are correct for part a.

For part b, note that $100\equiv-1\mod101$ so we can take differences of pairs of numbers $19-18+17-\ldots+11=15$ as the final answer.

answered Jun 12 '19 at 19:18

auscrypt

Thankyou so much! But i dont understand the part b, what is relation of finding mod =-1 and partition into two ? Can you explain a little bit more? – fiksx Jun 12 '19 at 19:57

sirous · Accepted Answer · 2019-06-13T06:40:18.240

0

$$N=19+18\times100+17\times100^2+16\times100^3+15\times100^4+14\times100^5+13\times100^6+12\times100^7+11\times100^8$$

a:$100≡1\ mod 99$ ⇒ $100^k≡1\mod 99$

⇒ $$N≡(11+12+13+14+15+16+17+18+19) mod 99 ≡135\ mod 99≡36 \ mod 99$$

b:$100≡ -1\ mod 101$ ⇒ $100^{2k}≡[(-1)^{2k}=1]\ mod 101$; $100^{2k+1}≡[(-1)^{2k+1}=-1]\ mod 101$

⇒ $$N≡(11-12+13-14+15-16+17-18+19) mod 101≡15\ mod 101$$

edited Jun 13 '19 at 06:40

answered Jun 12 '19 at 20:34

sirous

Thankyou but i want to know how you produce N SUCH AS 11-12+13...? What is the relation with the mod? And choose two digit only? – fiksx Jun 13 '19 at 04:08
@fiksx , I edited and showed why the signs change. – sirous Jun 13 '19 at 06:41
Is this mean to create partition for every two number that in base 10 implies $10^{2k}$ so when 11 it will be 10^16? – fiksx Jun 13 '19 at 06:45
@fiksx, I mean even number by 2k and odd number by 2k+1, i.e $(-1)^{even}=+1$ and $(-1)^{odd}=-1$ – sirous Jun 13 '19 at 07:53

2 Answers2