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Week 1 Homework assignment questions 3 and 9

 
 
DuongDuy的头像
Week 1 Homework assignment questions 3 and 9
DuongDuy - 2020年12月4日 Friday 17:49
 

Does anyone have any ideas/hints how to get started on these? 

 
ZhuDaniel的头像
Re: Week 1 Homework assignment questions 3 and 9
ZhuDaniel - 2020年12月4日 Friday 18:01
 

For #3 I would suggest getting a common denominator and using mod arithmetic from there.


I also need some help with #9.

DuongDuy的头像
Re: Week 1 Homework assignment questions 3 and 9
DuongDuy - 2020年12月4日 Friday 18:56
 

So I have (9a^2+14b^2)/b+9a

What do I do from here?

ZhuDaniel的头像
Re: Week 1 Homework assignment questions 3 and 9
ZhuDaniel - 2020年12月5日 Saturday 11:56
 

Try taking (mod a) and (mod b)

ZhangCharles的头像
Re: Week 1 Homework assignment questions 3 and 9
ZhangCharles - 2020年12月5日 Saturday 13:22
 

I actually got #9 correct (after thinking about it for hours and writing it on my whiteboard multiple times, and maybe writing a python program to do it for me)

My method was setting X and Y to be the base-5 and base-6 counterparts of N, respectively, and setting

N = abc (base 10)

X = defgh (base 5)

Y = jkmn (base 6)

Where the letters denote digits. (i.e. abc could be 243 or 398)


Then from here, remember that

X = 625d + 125e + 25f + 5g + h in decimal

Try converting all of them to decimal.  Remember N=X=Y in decimal!

Try getting an equation that relates c, h, and n, and then use the fact that the last 2 digits of S = the last 2 digits of N.


There are probably better solutions.

This is Charles Zhang, signing off for now.

DuongDuy的头像
Re: Week 1 Homework assignment questions 3 and 9
DuongDuy - 2020年12月5日 Saturday 14:52
 

What do you mean by decimal?

 By that I mean how does (N=X=Y)?


ZhangCharles的头像
Re: Week 1 Homework assignment questions 3 and 9
ZhangCharles - 2020年12月5日 Saturday 15:41
 

By decimal, I mean base 10 (you know, the base we all count in) 

N=X=Y because X and Y are just N written in base 5 and 6, respectively, and converting them back and forth into the same base (most useful would be base 10) will get you the same number.


Take the number in the problem given: 749

N = 749 (base 10)

X = 10444 (base 5)

Y = 3245 (base 6)

N=X=Y in base 10


However, by the way I set up variables, 

a=7, b=4, c=9

d=1, e=0, f=4, g=4, h=4

j=3, k=2, m=4, n=5

Then from here, there are some intermediate steps to get the answer.  (Such as equating them by what is given, being careful, and counting possibilities with casework.)



ZhangCharles的头像
Re: Week 1 Homework assignment questions 3 and 9
ZhangCharles - 2020年12月5日 Saturday 15:41
 

Hope this helped!  I am definitely not the best explainer.

ZhangCharles的头像
Re: Week 1 Homework assignment questions 3 and 9
ZhangCharles - 2020年12月5日 Saturday 15:42
 

I definitely took a while for me to understand it.  Sorry for not trying to help earlier.

DuongDuy的头像
Re: Week 1 Homework assignment questions 3 and 9
DuongDuy - 2020年12月6日 Sunday 09:07
 
Its fine, I figured it out after I looked at the explanation on AOPS. Thanks for the help, though.