Find the last two digits of the number (17)10.
49
We want to find the last two digits of the number (17)10. If we can express it in terms of a number which is a multiple of 10 or 100, it will be easy to find the last two digits. We know (17)2 = 289 = 290-1. We can start with this
(17)10=2895=(290−1)5
(290−1)5 = (5C0(290)5−5C1(290)4+5C2(290)3............+5C4(290)1−5C5)
(290−1)5 = (5C0(290)5−5C1(290)4+5C2(290)3............+5C4(290)1−1)
(290−1)5 = (A multiple of 1000) + 5(290)1−1
(290−1)5 = (A multiple of 1000) + 1449
⇒ The last three digits are 449