Find-the-last-digit-of-the-sum-19-81-4-9k-k-N-

Question Number 112996 by Aina Samuel Temidayo last updated on 10/Sep/20

Find the last digit of the sum

19^{81} + 4^{9 k}, k \in N

Answered by Aziztisffola last updated on 10/Sep/20

19^{81} + 4^{9 k} \equiv ? [10]

19 \equiv 9 \equiv - 1 [10] \Rightarrow 19^{81} \equiv {(- 1)}^{81} [10] \equiv - 1 [10]

4^{9} \equiv 4 [10] \Rightarrow 4^{9 k} \equiv 4^{k} [10]

\Rightarrow 19^{81} + 4^{9 k} \equiv 4^{k} - 1 [10]

k = 0 \Rightarrow last digit is 0

k = 1 \Rightarrow \dots . is 3

k = 2 \Rightarrow \dots . is 5

k = 3 \Rightarrow \dots . is 3

k = 4 \Rightarrow \dots . is 5

\dots . .

if k = even \Rightarrow last digit of 19^{81} + 4^{9 k}, k \in N

is 5 and if k = odd then last digit of 19^{81} + 4^{9 k}, k \in N

is 3 .

Commented by Aina Samuel Temidayo last updated on 10/Sep/20

Yea .