Question

The last digit of the number ((\(\sqrt{51}\) + 1)^51 – \(\sqrt{51}\) – 1)^51 is _______

(a) 32

(b) 8

(c) 51

(d) 1

This question was addressed to me in my homework.

This is a very interesting question from Counting topic in portion Counting of Discrete Mathematics

Answer 1

Correct option is (b) 8

To elaborate: Consider the binomial expansion of (m+1)^71 and (m-1)^71 which gives these two

expressions below respectively: 1) m^51 + ^51C1m^50 + ^51C2m^49 + ^51C3m^48 + … + ^51C50m^1 + ^51C51m^0

2) m^51 – ^51C1m^50 + ^51C2m^49 – ^51C3m^48 + … + ^51C50m^1 – ^51C51m^0 .

By taking the difference we have, 2(^51C1m^50 + ^51C3m^48 – ^51C5m^46 + … + ^51C50m^2 – ^51C51m^0 ).

In this case, m = \(\sqrt{51}\) and 2(^51C1m^50 + ^51C3m^48 – ^51C5m^46 + … + ^51C50m^2 – ^51C51m^0 ).

Consider, module 10 on the powers(for any natural number n): (51)^n ≡ (51 mod 10^n) ≡ 1 gives 2(^51C1 + ^51C3 + ^51C5 + … + ^51C50 + ^51C51). Now, by adding the odd terms of the 51^st row of the Pascal Triangle 2.(\(\frac{1}{2}\) * 2^51) = 2^51 = 2^(51 mod 4) = 2^3 = 8.