The number which is written as 1 followed by 20,000 zeroes
is divided by the number
which is written as 1 followed by 199 zeroes followed by a 7.
Find the last digit of the quotient (if divided with remainder).
[SECRET]
Q:zero를 입력하세요
A:zero
[solution]
Exponents are hard to write here,
so let us use a^b to mean "a to the power b".
Now divide the polynomial X^100 by the polynomial X+7;
then the quotient is
X^99 - 7 x^99 + 7^2 x^98 - ... + 7^98 X - 7^99
with remainder (-7)^100=7^100.
Now put X=10^200,
and since the remainder 7^100 is less than 10^200, no more division is needed.
So the digit we need is the last digit of (10^99 - 7^99).
Now 7^4-2401 ending with 1,
so 7^99 ends with the same digit as 7^3=343, since 99=4(24)+3.
So 10^99 - 7^99 ends with the digit 7.
[/SECRET]
'바로 콕 찍어서 - 진너자하 > 詩처럼아름다운數學' 카테고리의 다른 글
| [추천site] 입체도형 -http://www.uff.br/ (0) | 2009/05/14 |
|---|---|
| [학습도구Learning Tool] Screen Protractors (0) | 2009/05/13 |
| Dr.Math --the last digit -- problems of the week (0) | 2009/05/09 |
| [추천사이트] http://en.graphoon.org/ (0) | 2009/05/08 |
| [교육] 2008년 - 연세대학교 - 자연계열 - 논술 - 수학 - 본고사 (0) | 2007/11/27 |
| Dr.Math -- the minimum possible value -- problems of the week (0) | 2007/05/27 |
