Prove that if m is a natural number m(m+1) can never be a perfect power i.e. not a square or a cube etc. of a natural number.
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1 Answers
1m(m+1) = Kn say,
then
m(m+1)=k*k*k...n times
k can either be prime or composite.., in case of pime,
we have to split k*k*k... into 2 groups, 1 will be m, the other will be (m+1) which isnt possible, as m and 1+m are relatively coprime, when
any 2 groups from k*k*k... will have k as a common factor...
in case of composite, say k=apbq.... a,b..being primes,
we have m(1+m)=(apbq...)k
again we will have to divide (apbq...)(apbq...)(apbq...)... into 2 groups,
without a common factor, which is not possible
hence contradiction in both cases for k,
cheers!!
