Question

初等数论证明题

设n是任意正整数，α是实数，证明： [   [  nα ]/  n ]= [  α   ]有谁能解一下呢，谢谢！

Answer 1

α = [α] +{a}
nα =n[a] +n{a}
[nα] =n[a] +[n{a}]
[na]/n =[a] +[n{a}]/n
[ [nα]/n ] =[ [a] +[n{a}]/n ]
=[a] +[ [n{a}]/n ] --------n{a}<n 所以[n{a}]/n < n/n =1 所以　[ [n{a}]/n ]　＝0
＝[a] +0
　　　　＝[a]