What is the digit in the hundreds place of $5^{2017}$

Question

What is the digit in the hundreds place of $5^{2017}$?

Since $5^3 = 125$, powers with odd exponent of $5$, from the third onward, will end with the digits $125$, while those with even exponents will end with $625$. We can conclude that the digit of the hundreds of $5^{2017}$ is $1$. Is it correct or are there possible other solutions than mine?

Your solution is correct. Furthermore, it seems to me like your solution is the quickest & easiest way to solve this problem. — Robin, Dec 16 '23 at 13:13
Please, use descriptive titles. "Mathematical games 1" says nothing about the subject of the question. — jjagmath, Dec 16 '23 at 13:43

score 7 · Accepted Answer · edited Dec 16 '23 at 14:04

7

Yes, it is correct.

Let $n$ be an arbitrary positive integer greater than $1$.

Assume that $5^{2n-1}\equiv125\pmod {1000}$.

$$5^{2n+1}=5^{2n-1}\cdot25\equiv125\cdot25=3125\equiv125\pmod {1000}.$$

Thus it has been proven by induction that $5$ to the power of any odd number greater than $1$ will end with the digits $125$. The same can be done for the even powers.

edited Dec 16 '23 at 14:04

A rural reader

2,507

answered Dec 16 '23 at 12:46

Matthan

241

Thank you very much. However actually I not remember anything of Modular arithmetic. – Sebastiano Dec 16 '23 at 20:18

What is the digit in the hundreds place of $5^{2017}$

1 Answers1