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September 21st, 2014, 01:04 PM   #1
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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?
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September 21st, 2014, 01:50 PM   #2
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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?
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September 21st, 2014, 07:19 PM   #3
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Thank you very much
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January 31st, 2018, 05:51 PM   #4
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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.
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January 31st, 2018, 05:53 PM   #5
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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.
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January 31st, 2018, 06:00 PM   #6
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Quote:
Originally Posted by Tricia View Post
I have the same question expect I don't know why - the explanation which is required for my homework. Please assist.
You might have better luck making a new post of your own. By all means, if this thread is relevant, you can link it in your post.
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January 31st, 2018, 06:55 PM   #7
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Quote:
Originally Posted by Tricia View Post
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.
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.
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January 31st, 2018, 07:14 PM   #8
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Quote:
Originally Posted by Tricia View Post
I have the same question ....
Are you both in the same class?

I think the hard part of this problem is to figure out which ordinal position each letter is. I have to start at abc and count on my fingers.
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February 1st, 2018, 09:19 AM   #9
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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)
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February 1st, 2018, 09:26 AM   #10
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Quote:
Originally Posted by Tricia View Post
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.
"a ≡ x1 (mod m)" means a= x1+ km for some integer k.
"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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