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MCIII NT Chap1 HW help

 
 
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Re: MCIII NT Chap1 HW help
by Dr. Kevin Wang - Sunday, March 26, 2023, 12:32 AM
 

For 1.14, try to use Bezout's Identity.

For 1.15, if you have an equation, that's the way to go.  You can show your equation here and ask further. On the other hand, if you just try to calculate the digits one at a time, starting from the units digit, you can get it too, without using any equation.

For 1.16, try a few examples of $m$.  If you can solve the case where $m$ is a prime, then the non-prime case is not too different. Show how far you have got here, and you will get further hints.

For 1.17, try to add all values in a complete residue system. Again, an example can help you.

For 1.18: this is easier. No hint is needed.  Think a little more.

1.19 and 1.20 require some in-depth analysis.  For 1.19 you can try to consider cases where $n$ is even and $n$ is odd. 

For 2.20, start with the first digit.  What is $a$? Then think about the following questions: Can the digits repeat?  What can be the last digit? Narrow down the choices.

Picture of Katrina Liu
Re: MCIII NT Chap1 HW help
by Katrina Liu - Tuesday, April 11, 2023, 8:30 PM
 

For 1.15, my equation was: 2x10^a+N=20N+4, a being the number of digits of N. I got stuck there unfortunately.
For 1.16, I kind of have a solution but I'm not sure if it's correct.
if m is prime, then k and m-k are add to 0 modulo m, and there can't be one in the middle because that way 2 would divide m.
if m isn't, if a is part of its reduced residue system, then m-a must be as well. and we can pair them up.

Picture of Dr. Kevin Wang
Re: MCIII NT Chap1 HW help
by Dr. Kevin Wang - Wednesday, April 12, 2023, 12:02 PM
 

1.15 just need a little more thought. You can transform the equation to $2\times 10^a = 19N + 4$.  Think more carefully what this equation means. Hint: what do you do if you divide 200 by 19?  2000 by 19?  20000 by 19?

1.16 is correct.  Pairing is a good method that can be applied to many problems.  Good job!