The Pisano period modulo $n$ is the fundamental period of the repeating sequence of smallest positive integers that are congruent to the Fibonacci sequence modulo $n$ . It is known that the Pisano period modulo 10 is 60; thus, we only need to look at $1981 \pmod{60}$ .

Since 1980's digits sum to a multiple of 3, it is divisible by 3. It is clear 1980 has a factor of 20. Thus, 1980 is divisible by 60 and so $1981 = 1980 + 1 \equiv 1 \equiv 1 \pmod{60}$ . Therefore,

$F_{1981} \equiv F_{1} \equiv \boxed{1} \pmod{10}$

Using arithmatic sequence

a+(n-1)d= 1+(1980)*1,6...= .......1

So, the unit digit is 1

CMIIW