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What is an intuitive way to think about odd and even numbers? And about divisibility also...

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It is really vague question to ask. Could you explain yourself a bit better? –  Jack Yoon Apr 15 at 10:43
    
More intuitive than what they actually are? Even numbers = integers divisible by two, odd numbers = intergers not divisible by two... –  DonAntonio Apr 15 at 10:48
    
why is my question on hold? –  precalcII Apr 15 at 15:24

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up vote 4 down vote accepted

Think of indivisible bricks, and houses:

$\,\,\,\,$house1&2

You can cut the first house into two equal parts. But you're not strong enough to cut the second house into two equal part, since you'd be left with one brick.

For divisibility, build houses with bases containing $m$ bricks. Then the total number of full floors is the quotient of the total number of bricks, divided by $m$. And the remainder is the number of bricks in the floor where they aren't complete. Example with $9$ divided by $4$:

enter image description here

A number $m$ divides a number $t$ if and only if you can build a house with $t$ bricks with a base made of $m$ bricks and without having any incomplete floor.

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i love it ! very nice way! –  precalcII Apr 15 at 11:02
    
@anabayan Glad you liked it! $\overset{\cdot\cdot}\smile$ –  Hakim Apr 15 at 11:03

Even numbers are just numbers that can be cut in half. Or, another way, if you have an even number of objects, you can sort them in pairs of two, or you can split them into two equally sized groups. The odd numbers are those where you can't do that.

This is generalizable to other numbers, so $n$ is divisible by $k$ if you can sort $n$ objects into groups of size $k$ (or into $k$ equally sized groups)

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This seems even generalizable to arbitrary cardinalities :) Then every $\aleph_\alpha$ would be even. –  user2345215 Apr 15 at 10:47

$10111010_2$ is even because the units position is zero. If you slide the bits to the right one position, you have divided by $2$ and the number is now odd.

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