B = 1/1.2 + 1/3.4 +..+1/2015.2016
B = 1-1/2 + 1/3 - 1/4 +...+ 1/2015 - 1/2016
B = 1+ 1/2 + 1/3 +..+1/2015 + 1/2016 - 2( 1/2 + 1/4 + ..1/2016)
B = 1/1009 + 1/1010 +.. + 1/2016 ( dpcm)

\(A=\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{2015.2016}\)

\(A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{2015}-\frac{1}{2016}\)

\(A=\left(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+...+\frac{1}{2015}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{2016}\right)\)

\(A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2015}+\frac{1}{2016}\right)-2\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{2016}\right)\)

\(A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2015}+\frac{1}{2016}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{1008}\right)\)

\(A=\frac{1}{1009}+\frac{1}{1010}+\frac{1}{1011}+...+\frac{1}{2015}+\frac{1}{2016}\)

\(\Rightarrow B-A=\left(\frac{1}{1008}+\frac{1}{1009}+\frac{1}{1010}+...+\frac{1}{2016}\right)-\left(\frac{1}{1009}+\frac{1}{1010}+\frac{1}{1011}+...+\frac{1}{2016}\right)\)

\(\Rightarrow B-A=\frac{1}{1008}\)

\(CMR:\dfrac{1}{2}+\dfrac{1}{3}.4+\dfrac{1}{5}.6+...+\dfrac{1}{2015}.2016=\dfrac{1}{1009}+\dfrac{1}{1010}+...+\dfrac{1}{2016}\)

\(=\dfrac{1}{1.2}+\dfrac{1}{3.4}+...+\dfrac{1}{2015.2016}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2015}-\dfrac{1}{2016}\)

\(=1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2015}+\dfrac{1}{2016}-2\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2016}\right)\)

\(=\dfrac{1}{1009}+\dfrac{1}{1010}+...+\dfrac{1}{2016}\left(đpcm\right)\).

Đặt tổng là S

\(\Rightarrow S=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{2015}-\frac{1}{2016}\)

\(\Rightarrow S=\left(1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{2016}\right)-2\left(\frac{1}{2}+\frac{1}{4}+.....+\frac{1}{2016}\right)\)

\(\Rightarrow S=\left(1+\frac{1}{2}+....+\frac{1}{2016}\right)-\left(1+\frac{1}{2}+....+\frac{1}{1008}\right)\)

\(\Rightarrow S=\frac{1}{1009}+\frac{1}{1010}+....+\frac{1}{2016}\) (đpcm)

$2015=5.13.31$2015=5.13.31

Ta có: $1.2.....1007=1.2...5....13.....31...1007\text{ chia hết cho }5.13.31=2015$1.2.....1007=1.2...5....13.....31...1007 chia hết cho 5.13.31=2015

$1008.1009.....2004=1008....\left(1010\right)....\left(1014\right)...\left(1023\right)....2004$1008.1009.....2004=1008....(1010)....(1014)...(1023)....2004

$=1008....\left(5.202\right)....\left(13.78\right)....\left(31.33\right)...2004\text{ chia hết cho }5.13.33=2015$=1008....(5.202)....(13.78)....(31.33)...2004 chia hết cho 5.13.33=2015

Do đó tổng 2 số trên chia hết cho 2015.