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 Topic: modular arithmetic
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Calendar arithmetic 2016-02-14
From Jenalee:
January 1, 2001 is Julian Day 2 451 911 (the number of days that have passed since Day 0, January 1, 4713 BC).

If Julian Day 0 was a Monday, what day of the week was January 1, 2001?

Mod versus Rem in Turing 2013-01-01
From Eric:
I am a teacher teaching computer science using Turing. I am having difficulty understanding why one would use the mod operator versus the rem remainder operator.

Mod seems to make the resulting sign depend on the sign of the divisor, whereas rem makes the resulting sign depend on the dividend.

Examples:

11 mod 5 = 1 and 11 rem 5 =1
-11 mod 5 = 4 and -11 rem 5 = -1
11 mod -5 = -4 and 11 rem -5 =1
-11 mod -5 = -1 and -11 rem -5 = -1

What I can't understand is why this would matter. For example, -11 / 5 = -2.2 and 11 / -5 = -2.2 get the same result.
So how is a remainder dependent on the sign of one of the parts? What benefit would using one over the other have?

Any insight would be most helpful!

Eric

Modular arithmetic 2011-10-30
From Kim:
Hello,
I am editing a resource for students, and I think some of the answers may be incorrect. The text I was given and my questions are in the attachment. Any help you could give would be appreciated.
Thanks,

Kim

(x^3 + 11x) is divisible by 6 2010-06-24
From PT:
Given that x is a non-zero integer, how do you show that for all values of x, (x3 + 11x) is divisible by 6?

I know it works but how do I answer the "all values of x" part?

Three prime numbers p,q and r, all greater than 3, form an arithmetic progression: 2005-07-18
Three prime numbers p,q and r, all greater than 3, form an arithmetic progression: p=p, q=p+d and r= p+2d. Prove that d is divisible by 6.
Take It! 2002-04-03
From Bryan:
You are playing Take It! for \$180,00 with a total stranger. There are 180 identical balls in a big vase. Each player in his turn, reaches into the vase and pulls out 1,5,or8 balls. These balls are discarded. The player who takes the last ball from the vase wins the \$180,000. A flip of the coin determines that you will go first. Are you glad? How many will you take out on the first move, and how will you proceed to win the prize?
Finding a formula 2000-05-05
From Erica Hildebrandt:
If a farmer has a field and his plots are laid out in the following grid where each # represents a plot:
 4 5 12 13 20 3 6 11 14 19 2 7 10 15 18 1 8 9 16 17

Of course the plot numbers aren't meaningful as I have described above. In fact they may not be numbers at all. The only constants I have are the total number of rows and columns. Using the total number of rows and columns and my current position row and column, how can I write a formula that tells me column 3 row 3 = 10, column 4 row 2 = 14, etc. I can see the pattern but can't quite get the formula. I believe I will need 2 different formulas one for even and one for odd rows.
Answered by Paul Betts and Penny Nom.

Divisibility by 9 1999-02-21
From Razzi:
I've been having a hard time trying to solve the following problem and I was wondering if you could help me.

For any positive integer a let S(a) be the sum of its digits. Prove that a is divisible by 9 if and only if there exist a positive integer b such that S(a)=S(b)=S(a+b).
Answered by Chris Fisher and Harley Weston.

Modular Arithmetic 1999-02-04
From Leslie Kupper:
I am trying to do a project on modular arithmetic. I was wondering if there were any websites that include a sample lesson plan on modular arithmetic for any grade level. Let me know where and how to find them. Thanks.