×

# Help: Mathematics UIL Question

I am trying to get a head start on practicing for UIL mathematics, and I am stumped on this random practice problem. I have no clue how to solve it. The question is:

What is the sum of the digits in the tens place and the units place of 7^65

I don't just want the answer, I have that, I want to be able to solve other questions like this. I appreciate the help.

Note by Alder Fulton
4 months, 2 weeks ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

Relevant article: finding the last few digits of a power

- 4 months, 2 weeks ago

You can notice that the unit digit change properly:

$$\color{red}\dots 7, \color{blue}\dots 9, \color{green}\dots 3, \color{yellow}\dots 1,\color{red}\dots 7, \color{blue}\dots 9, \color{green}\dots 3, \color{yellow}\dots 1,\color{red}\dots 7, \color{blue}\dots 9, \color{green}\dots 3, \color{yellow}\dots 1,\color{black}\dots$$

Since $65\equiv 1 \ \mod 4$, the unit digit of the $$7^{65}$$ will be 7.

You can notice that the tens digit change properly:

$$\color{pink}\dots0., \color{green}\dots4., \ \dots4., \color{pink}\dots0., \ \dots0., \color{green}\dots4., \ \dots4., \color{pink}\dots0., \ \dots0., \color{green}\dots4., \ \dots4., \color{pink}\dots0., \ \dots0.,\color{black}\dots$$

So the first is 0, then 4400 repeat. So we subtract 1 from 65 (since the first, alone 0), and we know that

$64\equiv0 \ \mod4$

so the tens digit of $$7^{65}$$ will be 0 (the last one from the repeating sequence).

So the sum is 7.

Do you need proof for the notices, or it is enough?

- 4 months, 2 weeks ago

Thank you so much! This helped a lot.

- 4 months, 2 weeks ago