+0  
 
0
111
3
avatar+162 

What is the residue of 9^2010, modulo 17?

 

How many zeroes does 10! end in, when written in base 11?

 

Let $n$ be the integer such that $0 \le n < 31$ and $3n \equiv 1 \pmod{31}$. What is $\left(2^n\right)^3 - 2 \pmod{31}$? Express your answer as an integer from $0$ to $30$, inclusive.

 

What is the sum of all positive integer solutions less than or equal to $20$ to the congruence $13(3x-2)\equiv 26\pmod 8$?

Creeperhissboom  May 27, 2018
 #1
avatar
+1

9^2010 mod 17 = 13

 

10! base 10 =3,628,800 =205940a in base 11. So, there are no trailing zeros in base 11.

 

Sorry, can't read the other 2.

Guest May 27, 2018
 #2
avatar+901 
+2

What is the residue of \(9^{2010}\), modulo 17?

 

We can see the pattern:

 

\(9^1\equiv9\pmod{17}\\ 9^2\equiv13\pmod{17}\\ 9^3\equiv15\pmod{17}\\ 9^4\equiv16\pmod{17}\\ 9^5\equiv8\pmod{17}\\ 9^6\equiv4\pmod{17}\\ 9^7\equiv2\pmod{17}\\ 9^8\equiv1\pmod{17}\\\)

The residues then repeat after that. 

 

Therefore, since \(2010รท8=251\ R\ 2\), the residue is 13. 

 

I hope this helped,

 

Gavin

GYanggg  May 27, 2018
 #3
avatar+901 
+2

Let n be the integer such that \(0\le n < 31\ \text{and}\ 3n \equiv 1 \pmod{31}.\) What is \(\left(2^n\right)^3 - 2 \pmod{31}\)? Express your answer as an integer from 0 to 30, inclusive.

 

\(3n\equiv1\pmod{31}\\ 3n\cdot21\equiv1\cdot21\pmod{31}\\ 63n\equiv21\pmod{31}\\ 62n+n\equiv21\pmod{31}\\ \boxed{n\equiv21\pmod{31}} \)

 

If \(0\le n < 31\ \text{and}\ 3n \equiv 1 \pmod{31},\) n = 21

 

You can finish this on your own. 

 

What is the sum of all positive integer solutions less than or equal to 20 to the congruence \(13(3x-2)\equiv 26\pmod 8\)?

 

Note: you usually can not divide in linear congruences, but since 13 is relatively prime to 8, you can. 

 

\(13(3x-2)\equiv 26\pmod 8\\ 3x-2\equiv2\pmod8\\ 3x\equiv4\pmod8\\ 3x\cdot3\equiv4\cdot3\pmod8\\ 9x\equiv12\pmod8\\ 8x+x\equiv12\pmod8\\ x\equiv12\pmod8 \)

 

I hope this helped,


Gavin

GYanggg  May 27, 2018
edited by GYanggg  May 27, 2018

20 Online Users

avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.