hmm
Ill try to keep alive this section
I guess a lot of different ppl come to this thread, so here is one constructive in nature that can (supposedly) be enjoyed by ppl with different degrees of knowledge; it also serves as basis to expand ur knowledge of math while we r at it (if u didnt know about it already)!
To solve this u will need basic algebra (think about working with the x and decomposing a number in prime factors) and know about sum and multiplication.
So first lets begin with an operation:
Module.
We will write the module operation between 2 numbers x and y with the symbol % (x%y) and read it as "x module y".
The intuition behind module is the remainder of an integer division; 9%2=1, for example, because 9/2= 8/2+1/2; 8/2=4, but 1 is left as a remainder, as it can not be divided by 2 into an integer number.
Formally, we say that p%q=r if and only if there exists an integer n such as p=n*q+r, with r>=0 and r<q.
U can prove it if u want, but it is not necessary, that the operation is well defined and whatever formality nitpicks u have.
Modules generalize a lot of the concepts we work with, and we use that operation constantly, even if we dnt realize it. Want an example? Lets say it is 5:30 Pm; wich minute well have in 92 minutes? Those quick at calculating stuff will now it will be 7:02; it is 60+30+2 +the initial 30, rendering 60+60+2. The answer to wich minute is it is the answer to (30+92)%60; with hours, it works the same way. if it is 23:00, in 48 hours it will be 23:00, and in 5 hours it will be 4:00; that is a module by 24.
so, lets go to the challenges:
1. Prove that % is distributed over addition, that is:
(x+y)%z=(x%z+y%z)%z
Examples:
(5%3+11%3)%3=(2+2)%3=1=(5+11)%3
Hint:
2. Prove that the sum of 2 numbers is even if and only if both numers are even or both numbers are odd
Hint:
3. Prove that the same thing above occurs with multiplication:
(x*y)%z=(x%z*y%z)%z
Example:
(7%3)*(11%3)%3=(1*2)%3=2=(7*11)%3=77%3 (75=3*25)
Hint:
Now comes the FUN stuff! Well, if u r stilll with me it must mean u r having fun, right? Right? Well, if u r:
Remember those tricks we used when we were learning the multiplicaction table and what nots? now well prove some of them:
4. Prove that a number is a multiple of 2 if the last digit is a multiple of 2, and that a number is multiple of 5 if the last digit is 0 or 5.
Hint:
5. Prove that a number is multiple of 3 if the sum of its digits is a multiple of 3
Hint:
6. What is the last digit of 7^49?
Hint:
So there it is! The idea was to be doable by a lot of ppl (hopefully), and at the same time answering some of the mysteries we might have had on the school. Remember to spoiler ur answers, and leave more puzzles/exercises/challenges!