November 28th, 2017, 07:16 PM |
#1 |

Newbie Joined: Sep 2017 From: San Diego Posts: 8 Thanks: 0 | What does this mean
If the GCD(a, m) =1, show that any x such that $\displaystyle x \equiv ca^{\phi(m)-1}\mod m $ satisfies $\displaystyle ax\equiv c\mod m$ I put Suppose $\displaystyle x \equiv ca^{\phi(m)-1}\mod m $ for any x. Then by multiplying both side of congruence by a we get, $\displaystyle ax \equiv ca^{\phi(m)}\mod m $ Since the GCD(a, m) =1, by Euler's theorem it follows that $\displaystyle a^{\phi(m)} = 1\mod m $ Thus, $\displaystyle ax \equiv ca^{\phi(m)}\equiv c \cdot 1 \mod m $ Hence $\displaystyle ax \equiv c \mod m $ Is this what I need to do? To me the word satisfies means is a solution to, but I have no idea how to show that. This all I came up with. |

December 1st, 2017, 03:58 AM |
#2 |

Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 | |