X  
View & Edit Profile Sign out
X
Sign in
Google
Facebook
Twitter
Yahoo
LinkedIn
X
Discussion Board
showing 1-2 of 2 (answers : 2, comments : 0),   sorted newest to the oldest
Question
when $13^{73}+14^{3}$ is divided by $11$ then remainder is
 
(1) (0) Comment Answer
numbers
2016-12-19 18:35:19 
gajanand
2 Answers
We know that 

Suppose $a$ is an integer and $p$ is prime. Then, according to Fermat's little theorem,
remainder$\left(a^{p-1}/p\right)=1$

Therefore,
remainder$\left(13^{11-1}/11\right)=1$

It means
remainder$\left[\left(13^{10}\right)^7/11\right]=1$
$\Rightarrow$ remainder$\left(13^{70}/11\right)=1$

So we have to solve now
$13^3/11+14^3/11$

$13^3=2197\\
14^3=2744$

We can write here
$(2197+2744)/11\\
4941/11$

Finally remainder we got
Remainder $=2$

Answer is $2$
 
(0) (0) Comment
2016-12-24 13:22:45 
lakhan
Ans: $2$

$13^{73}\equiv (11+2)^{73}\equiv 2^{73} \\ \equiv \left(2^5\right)^{14}×2^3 \equiv (32)^{14}×2^3 \\
\equiv (-1)^{14}×8 \equiv 8 \pmod{11} ~~\cdots(a)$

$14^{3}\equiv (11+3)^{3}\equiv 3^{3} \\ \equiv 5  \pmod{11} ~~\cdots(b)$

From $(a)$ and $(b)$
$13^{73}+14^3 \equiv (8+5) \equiv 13 \equiv 2 \pmod{11}$

Note: click here to understand about congruence relation.
 
(0) (0) Comment
2016-12-19 19:18:52 
Jay Patel (Senior Maths Expert, careerbless.com)
Answer this Question

Name
0 + 1 = (please answer the simple math question)
Post Your Answer