Power of 4: Last Digit Analysis

  • Thread starter Thread starter dainty77
  • Start date Start date
  • Tags Tags
    Analysis Power
Click For Summary

Homework Help Overview

The discussion revolves around analyzing the last digit of n^4, where n is a natural number. Participants explore the potential last digits of n^4, specifically focusing on the digits zero, one, five, or six.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants consider various methods, including case analysis and direct proof. Questions arise about the effectiveness of these approaches and the reasoning behind them.

Discussion Status

The discussion is active, with participants sharing different strategies and questioning the validity of their approaches. Some are exploring the implications of odd and even numbers in their reasoning.

Contextual Notes

There is an emphasis on understanding the relationship between n and its fourth power, particularly regarding how the last digit is determined. Participants are also considering the implications of different representations of n.

dainty77
Messages
9
Reaction score
0

Homework Statement



If n is a natural number, then n^4 ends in either zero, one, five, or six.

Homework Equations





The Attempt at a Solution



Should I attempt this by cases?
 
Physics news on Phys.org
dainty77 said:
Should I attempt this by cases?

What cases did you have in mind? Why don't you show us what you are thinking.
 
Actually not cases, but by a direct proof so:

let n^4=(n^2)^2
Let n be an odd number
Then n=2k+1 for some integer k
then n^2= (2k+1)^2
=4k^2 + 4k +1
=2(2k^2+2k) + 1

I don't think this is proving anything. I will try something else
 
Let n = 10m + k

Where m and k are naturals and k lies in the interval [0,9].

Take the 4th power of this expression. Can you find the term responsible for the final digit of n^4? Why is it responsible for the final digit? What are its possible values?
 
Last edited:

Similar threads

Finding the Last Non-Zero Digit of a Repeated Factorial Expression
  • · Replies 1 ·
Replies
1
Views
1K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
  • · Replies 3 ·
Replies
3
Views
2K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
First multiple of 2019 with last digits 2018
  • · Replies 4 ·
Replies
4
Views
2K
Fourier series of translated function
  • · Replies 1 ·
Replies
1
Views
2K
What is the radius of convergence for a series with logarithmic terms?
  • · Replies 3 ·
Replies
3
Views
2K
Intervals of Convergence- Power Series
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad What is the probability that the product of the digits is odd?
  • · Replies 13 ·
Replies
13
Views
3K
Finding last two digits for a large exponentiation
  • · Replies 3 ·
Replies
3
Views
2K
Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits
  • · Replies 32 ·
2
Replies
32
Views
4K