CMR
\(2008^{100}+2008^{99}⋮2009\)
c/m : 2008^100 + 2008^99 chia hết cho 2009
Ta có:
2008100+200899
= 200899.2008+200899
= 200899.(2008+1)
= 200899.2009
Vì 2009 chia hết cho 2009 => 200899.2009 chia hết cho 2009 => 2008100+200899 chia hết cho 2009
2008100 + 200899 = 200899(2008+1) = 200899. 2009
=> 2008100 + 200899 chia hết cho 2009
Chứng minh rằng 2008100+200899⋮2009
Ta có :
\(2008^{100}+2008^{99}\)
\(=2008^{99}.\left(2008+1\right)\)
\(=2008^{99}.2009⋮2009\)
=> đpcm
Học tốt
Bài làm :
Ta có :
\(2008^{100}+2008^{99}=2008^{99}.\left(2008+1\right)=2008^{99}.2009⋮2009\)
=> Điều phải chứng minh
\(2008^{100}+2008^{99}\)
\(=2008^{99}\left(2008+1\right)\)
\(=2008^{99}\cdot2009⋮2009\)
c/m : 2008100 + 200899 chia het cho 2009
CM 2008100+200899 chia hết cho 2009
\(2008^{99}\cdot2008+2008^{99}\)
\(=2008^{99}\cdot\left(2008+1\right)\)
\(=2008^{99}.2009⋮2009\left(dpcm\right)\)
Ta có: 2008100+200899=200899.2008+200899.1
=200899.(2008+1)
=200899.2009
=200899.2009 \(⋮\)2009
\(2008^{100}+2008^{99}=2008^{99}.\left(2008+1\right)=2008^{99}.2009⋮2009\) ( đpcm )
\(2008^{100}+2008^{99}\)
\(=2008^{99}.\left(2008+1\right)\)
\(=2008^{99}.2009⋮2009\)
=> đpcm
1.A=(2/3+3/4+4/5+................+99/100)*(1/2+2/3+..............+98/99);B=(1/2+2/3+..............+99/100)*(2/3+3//4+...................+98/99)
Tính A và B bằng cách thuận tiện nhất.
2.Cho a=2008/2009;b=2009/2008;c=1/2009;d=2007/2008
Tính a-b+c+d
3.Tìm STN m biết:
2016+m/m+2520+m/m+3024+m/m
So Sánh
a,A= \(\frac{2008^{2008}+1}{2008^{2009}+1}\)và B=\(\frac{2008^{2007}+1}{2008^{2008}+1}\)
b, M=\(\frac{100^{100}+1}{100^{99}+1}\)và N= \(\frac{100^{101}+1}{100^{100}+1}\)
a) Áp dụng \(\frac{a}{b}< 1\Leftrightarrow\frac{a}{b}< \frac{a+m}{b+m}\) (a;b;m \(\in\) N*)
Ta có:
\(A=\frac{2008^{2008}+1}{2008^{2009}+1}< \frac{2008^{2008}+1+2007}{2009^{2009}+1+2007}\)
\(A< \frac{2008^{2008}+2008}{2008^{2009}+2008}\)
\(A< \frac{2008.\left(2008^{2007}+1\right)}{2008.\left(2008^{2008}+1\right)}=\frac{2008^{2007}+1}{2008^{2008}+1}=B\)
=> A < B
b) Áp dụng \(\frac{a}{b}>1\Leftrightarrow\frac{a}{b}>\frac{a+m}{b+m}\) (a;b;m \(\in\) N*)
Ta có:
\(N=\frac{100^{101}+1}{100^{100}+1}>\frac{100^{101}+1+99}{100^{100}+1+99}\)
\(N>\frac{100^{101}+100}{100^{100}+100}\)
\(N>\frac{100.\left(100^{100}+1\right)}{100.\left(100^{99}+1\right)}=\frac{100^{100}+1}{100^{99}+1}=M\)
=> M > N
CMR: \(\dfrac{2009}{\sqrt{2008}}+\dfrac{2008}{\sqrt{2009}}>\sqrt{2008}+\sqrt{2009}\)
vế trái = \(\dfrac{2008+1}{\sqrt{2008}}+\dfrac{2009-1}{\sqrt{2009}}=\sqrt{2008}+\sqrt{2009}+\dfrac{1}{\sqrt{2008}}-\dfrac{1}{\sqrt{2009}}\)
vì \(\dfrac{1}{\sqrt{2008}}-\dfrac{1}{\sqrt{2009}}>0\) nên suy ra đpcm
chứng minh rằng
a, 2008 ^100 +2008^99 chia hết cho 2009
b, 12345^678 -12345^677 chia hết cho 12344
a) Ta có:
\(2008^{100}+2008^{99}\)
\(=2008^{99}.\left(2008+1\right)\)
\(=2008^{99}.2009\)
Vì \(2009⋮2009\) nên \(2008^{99}.2009⋮2009.\)
\(\Rightarrow2008^{100}+2008^{99}⋮2009.\)
b) Ta có:
\(12345^{678}-12345^{677}\)
\(=12345^{677}.\left(12345-1\right)\)
\(=12345^{677}.12344\)
Vì \(12344⋮12344\) nên \(12345^{677}.12344⋮12344.\)
\(\Rightarrow12345^{678}-12345^{677}⋮12344.\)
Chúc bạn học tốt!