Question

the number in units place in 2¹⁹⁶⁷​

Answer 1

Answer:

8

Step-by-step explanation:

1967 = 4(mod3)

so unit digit of 2^1967 will be 8

Answer 2

We know that, for every $n\in\mathbb{N},$

$\longrightarrow 6^n\equiv 6\pmod{10}\quad\quad\dots(i)$

We see that,

$\longrightarrow2^4=16\equiv6\pmod{10}$

$\longrightarrow2^4\equiv6\pmod{10}\quad\quad\dots(1)$

Now we find the quotient obtained on dividing 1967 by 4.

$\longrightarrow\left[\dfrac{1967}{4}\right]=491$

[Note: [.] represents greater integer function.]

Then, raising each term in (1) to the power 491,

$\longrightarrow(2^4)^{491}\equiv6^{491}\pmod{10}$

By (i),

$\longrightarrow2^{1964}\equiv6\pmod{10}$

Multiplying by $2^3=8,$

$\longrightarrow2^{1964}\times2^3\equiv6\times8\pmod{10}$

$\longrightarrow2^{1967}\equiv48\pmod{10}$

As $48\equiv8\pmod{10},$

$\longrightarrow\underline{\underline{2^{1967}\equiv\mathbf{8}\pmod{10}}}$

Hence 8 is the answer.