Unit place

Number Theory Level 2

What will be the digit at the unit place of the following: $4127^{8329}$ ?

7 3 1 9

1 solution

Tan T
Apr 5, 2016

so this can be analysed as follows;
the base in the exponent given is 4127,right?
its clear that the digit at the units place of (4127)^1 is 7
further the digit at the units place of (4127)^2 is 9 (as units place will be the units place of 7X7=49 )
similarly the digit at the units place of (4127)^3 is 3
and the digit at the units place of (4127)^4 is 1
after the fourth power the digit at units place will repeat themselves in the order of 7 ,9,3,1,7,9,3,1 and so on you can cross check
so the units place digit repeats itself after 4 powers
hence we divide the power given i.e.8329 by 4 and the remainder will either be 1,2,3 or 4
if its 1 then units place will be 7
if its 2 then units place will be 9
if its 3 then units place will be 3
if its 4 then units place will be 1
as we inferred above.
this can be done with any power and any base you just have to find after how many powers the units place will start repeating. hope that helps ! :)

Excellently written! +1. Here's another way to do it using modular arithmetic :

We know that $4127\equiv 7\pmod{10}$ and, $7^2\equiv 9\equiv -1\pmod{10}$ , thus: $\begin{aligned} 4127^{8329}&\equiv 7^{8329}&\pmod{10}\\ &\equiv (7^2)^{4164}\times 7\\ &\equiv 1\times 7\\ &\equiv 7 \end{aligned}$

Thus the remainder is $7$ .

Sravanth C. - 5 years, 2 months ago

oh! thank you sravanth. I didn't really know the modular arithmetic part. thanks for introducing it to me.

Tan T - 5 years, 2 months ago

Unit place

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