September 21st, 2014, 01:04 PM  #1 
Newbie Joined: Sep 2014 From: Canada Posts: 4 Thanks: 0  Modulos Please help me solve this problem Problem 1 Let: (A=1, B=2, C=3,.., Z=26) a) What is (A + N) mod 26 in this system? b) What is (B + 6) mod 26 in this system? c) What is (Y  4) mod 26 in this system? d) What is (C  9) mod 26 in this system? Problem 2 a) The remainder of any positive integer when divided by 100 is the integer made of the two rightmost digits. b) The remainder of any positive integer when divided by 1000 is the integer made of the three rightmost digits. c) If a ≡ x1 (mod m) and b ≡ y1 (mod m), then we will have ab ≡ x1y1 (mod m). True or False? 
September 21st, 2014, 01:50 PM  #2 
Math Team Joined: Apr 2010 Posts: 2,778 Thanks: 361 
1.) a.) A + N = 1 + 14 ≡ 15 mod 26 in this system. b.) To what value is B equal to? c.) To what value is Y equal to? d.) To what value is C equal to? You may want to add 26. 2.) a.) Any integer is of the form 100 * k + r where k is any integer and r are the two rightmost digits. b.) Can you do something similar as for a.)? c.) Let a = k * m + x1 and Let b = l * m + y1. Can you compute a * b mod m? 
September 21st, 2014, 07:19 PM  #3 
Newbie Joined: Sep 2014 From: Canada Posts: 4 Thanks: 0 
Thank you very much

January 31st, 2018, 05:51 PM  #4 
Newbie Joined: Jan 2018 From: Toronto Posts: 12 Thanks: 0  Similar questions  Remainder of a positive integer when divided by 100
I have the same question expect I don't know why  the explanation which is required for my homework. Please assist. The remainder of any positive integer when divided by 100 is the integer made of the two rightmost digits. True or False and why?  I answered True, but I don't know why. I have been searching online to understand the concept of why it is made of two rightmost dights. 
January 31st, 2018, 05:53 PM  #5 
Newbie Joined: Jan 2018 From: Toronto Posts: 12 Thanks: 0 
Also, I don't understand this question  Please help me understand why it is true or false. If a ≡ x1 (mod m) and b ≡ y1 (mod m), then we will have ab ≡ x1y1 (mod m). True or False and why? Thank you. 
January 31st, 2018, 06:00 PM  #6 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,730 Thanks: 602 Math Focus: Yet to find out.  
January 31st, 2018, 06:55 PM  #7 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 619 Thanks: 83  If I understand, it's saying x and y are both multiples of m, which makes xy a multiple of m, which makes the statement true.

January 31st, 2018, 07:14 PM  #8 
Senior Member Joined: Aug 2012 Posts: 2,075 Thanks: 593  
February 1st, 2018, 09:19 AM  #9 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
Since A= 1 and Z= 26, by "counting on my fingers" (literally!) N= 14. (a) What is (A + N) mod 26 in this system? Whatever number "N" is in this system. Since A= 1. "A+ N" is the very next letter, "O". b) What is (B + 6) mod 26 in this system? Counting forward from B six places, C, D, E, F, G, H. (B+ 6)= H (mod 26) c) What is (Y  4) mod 26 in this system? Going back from Y four places, X, W, V, U. (Y 4)= U (mod 26) d) What is (C  9) mod 26 in this system? "mod" is cyclic. 3 9= 6= 26 6= 20 (mod 26) and the 20th letter is T. Or, starting from C and going back 9 places, B, A, Z, Y, X, W, V, U, T. C 9= T (mod 26) 
February 1st, 2018, 09:26 AM  #10  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894  Quote:
"b ≡ y1 (mod m)" means b= y1+ pm for some integer p. So ab= (x1+ km)(y1+ pm)= x1y1+ x1pm+ kmy1+ kpm^2= x1y1+ m(x1+ ky1+ kpm) which is "x1y1+ m times an integer" so is x1y1 (mod m).  

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b) if a ≡ x1 (mod m) and b ≡ y1 (mod m), then we will have ab ≡ x1y1 (mod m),if a ≡ x1 (mod m) and b ≡ y1 (mod m), then we will have ab ≡ x1y1 (mod m)
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