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.

#NumberTheory

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

You can notice that the unit digit change properly:

$\color{#D61F06}\dots 7, \color{#3D99F6}\dots 9, \color{#20A900}\dots 3, \color{#CEBB00}\dots 1,\color{#D61F06}\dots 7, \color{#3D99F6}\dots 9, \color{#20A900}\dots 3, \color{#CEBB00}\dots 1,\color{#D61F06}\dots 7, \color{#3D99F6}\dots 9, \color{#20A900}\dots 3, \color{#CEBB00}\dots 1,\color{#333333}\dots$

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:

$\color{#E81990}\dots0., \color{#20A900}\dots4., \ \dots4., \color{#E81990}\dots0., \ \dots0., \color{#20A900}\dots4., \ \dots4., \color{#E81990}\dots0., \ \dots0., \color{#20A900}\dots4., \ \dots4., \color{#E81990}\dots0., \ \dots0.,\color{#333333}\dots$

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?

Áron Bán-Szabó - 3 years, 11 months ago

Thank you so much! This helped a lot.

alder fulton - 3 years, 11 months ago

Relevant article: finding the last few digits of a power

Pi Han Goh - 3 years, 11 months ago

Math	Appears 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}`	$\boxed{123}$