Question

find the last two digits of 3^2012 when represented in decimal notation

Answer 1

[tex]3^{2012} = 9^{1006}=(10-1)^{1006} \\ \\ By\: using\: binomial\: expansion \\ \\ 10^{1006}-1006\times10^{1005}+.........+505515\times 10^{2}-1006\times 10+1 \\ \\ =10^{1006}-1006\times10^{1005}+.........+505515\times 10^{2}-10060+1 \\ \\ When\: this\: is\: divided\: by\:100\:\:we\: will\: get\: a\: remainder\:41 \\ \\ So\:\: last\: two\:digits\: of\:the\: expansion\:will\:be\:\:41[/tex]