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avatar+47 

1. Find all integers, \(0 \le n < 163\) such that n is its own inverse modulo 163

 

2. The inverse of a modulo 44  is b. What is the inverse of 9a modulo 44  in terms of b?

 

3. Let x and y be integers. If x and y satisfy, then find the residue of x modulo 72.

 

These problems are tough for me so any help would be appreciated...

 Aug 18, 2024
 #1
avatar+390 
-3

Here is the solution to problem 2:

 

Given that \( b \) is the inverse of \( a \) modulo 44, we have the relationship:

 

\[
ab \equiv 1 \pmod{44}
\]

 

We need to find the inverse of \( 9a \) modulo 44 in terms of \( b \). Let the inverse of \( 9a \) modulo 44 be \( x \). This means:

 

\[
(9a)x \equiv 1 \pmod{44}
\]

 

We can factor out the \( a \):

 

\[
9(ax) \equiv 1 \pmod{44}
\]

 

Since \( ab \equiv 1 \pmod{44} \), we replace \( ax \) with \( b \):

 

\[
9b \equiv 1 \pmod{44}
\]

 

Thus, \( x = 9b \) is the inverse of \( 9a \) modulo 44.

 

So, the inverse of \( 9a \) modulo 44 in terms of \( b \) is \( \boxed{9b} \).

 Aug 18, 2024

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