Although some you might have seen this question before give it a try.
If p,q are positive integers such that-
1- 12 + 13 - 14+.........- 11318 + - 11319 = pq
Show that 1979 divides p.
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1 Answers
1708\hspace{-16}\bf{\Rightarrow 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.........-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}}$\\\\\\ $\bf{\Rightarrow \left(1+\frac{1}{2}+\frac{1}{3}+.........+\frac{1}{1319}\right)-2.\left(\frac{1}{2}+\frac{1}{4}+.........+\frac{1}{1318}\right)=\frac{p}{q}}$\\\\\\ $\bf{\Rightarrow \left(1+\frac{1}{2}+\frac{1}{3}+.........+\frac{1}{1319}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+.........+\frac{1}{659}\right)=\frac{p}{q}}$\\\\\\ $\bf{\Rightarrow \frac{1}{660}+\frac{1}{661}+............................+\frac{1}{1319}=\frac{p}{q}}$\\\\\\ $\bf{\Rightarrow \left(\frac{1}{660}+\frac{1}{1319}\right)+\left(\frac{1}{661}+\frac{1}{1318}\right)+.......+\left(\frac{1}{979}+\frac{1}{990}\right)=\frac{p}{q}}$\\\\\\ $\bf{\Rightarrow 1979.\left(\frac{1}{660.1319}+\frac{1}{661.1318}+......+\frac{1}{989.990}\right)=\frac{p}{q}}$\\\\\\
\hspace{-16}$Now Here $\bf{1979}$ is a prime no.\\\\\\ So $\bf{p}$ is a multiple of $\bf{1979}$\\\\\\ or $\bf{p}$ is divisible by $\bf{1979}$
Shaswata Roy Nice.:)Upvote·0· Reply ·2013-02-24 04:22:33