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Number Theory I-B

 
 
LinAmber的头像
Number Theory I-B
LinAmber - 2019年08月20日 Tuesday 22:29
 

For the following questions in Chapter 9, I don't know how to solve them:

9.22

9.25

9.29.

 
ReynosoDavid的头像
Re: Number Theory I-B
ReynosoDavid - 2019年08月21日 Wednesday 10:53
 

For 9.22 and 9.25 try to find a pattern by looking at smaller powers of the numbers. For example, The first few powers of $2$ are $2, 4, 8, 16, 32, 64, 132, \dots$ and their remainders when dividing by $15$ are $2, 4, 8, 1, 2, 4, 8,\dots$, so we can see they repeat every $4$ powers; since $102$ has reminder $2$ when dividing by $4$, the remainder of $2^{102}$ when dividing by $15$ will be the $2^\text{nd}$ remainder on the list, so $4$.

For 9.29: Recall $a \equiv b \pmod{m}$ if $m$ divides $b - a$. So, the possible values of $m$ are numbers that divide the difference $1023 - 561$.