+38 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...
+310 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 \( \boxed{9b} \).
