How do I know by looking at this number (& using simple math) that it is evenly divisible by these numbers?
90506
I'll take it as far as I can.\n\n3 - Add all the numbers and if the result is divisible by 3 then the entire number is 9+1+2+0+3+1+2=18 which is divisible by 3.\n4 - The last 2 numbers are divisible by 4. 12 is divisible by 4 so the entire number is.\n6 - It's divisible by 2 and 3.\n8 - If the last 3 numbers are divisible by 8, then the entire number is. 312/8=39. \n9 - Like for 3, except it has to be divisible by 9. The result, from above, is 18 which is divisible by 9.\n12 - If divisible by 3 and 4.\n18 - If divisible by 2 and 9.\n24 - If divisible by 8 and 3.\n36 - If divisible by 4 and 9.\n72 - If divisible by 8 and 9.
1075517
There are actually a lot of divisibility tests for the number 7.  I'll prove the one that's been mentioned the most so far in the answers above, which is: "take off the last digit, double it, and subtract that result from the rest of the number."\n\nFor instance: is 1064 divisible by 7?  Well:\n\n106 - (2*4) = 106-8 = 98.  But is 98 divisible by 7?\n\n9 - (2*8) = 9-16 = -7, which is divisible by 7.\n\nBut why does it work?\n\nWell, consider that any number of more than two digits can be written as 10a + b, where b is the last digit and a is the number formed from the rest of the digits.  (For example, 1064 = 10*106 + 4.  a=106, b=4).\n\nThe "re-formed" number, then, when we subtract double the last digit, is a - 2b.  With me so far?\n\nNow, if a-2b is a multiple of 7, then we can write it as 7k, where k is some integer:\n\na - 2b = 7k\n\nNow add 49a+7b to both sides:\n\na - 2b + 49a + 7b = 7k + 49a + 7b\n\nSimplifying the left side:\n\n50a + 5b = 7k + 49a + 7b\n\nFactoring a 5 from the left side, and a 7 from the right:\n\n5(10a + b) = 7(k + 7a + b)\n\nNow, a, b, & k are all integers, so k+7a+b is too.  That means the right side is 7 times some integer -- in other words, a multiple of 7.\n\nBut the left side is 5 times our original number (10a+b), and since 5 isn't a multiple of 7, 10a+b must be.\n\nThat proves, then, that as long as a-2b is a multiple of 7, so is 10a+b.\n\nHope that helps!
812323
The prime factors of 1992 are 2 x 2 x 2 x 3 x 83.\n\nBy including 2 * 4 * 6, you have already covered 2 x 2 x 2 x 3, but you need to cover 83.  83 is prime, so the first time you have an even number that has 83 as a factor will be 166 (2 * 83).\n\nThat means you must figure the product of 2 * 4 * 6 * ... * 164 * 166 for it to be divisible by 1992.\n\nThe answer is largest even number in the product is 166.
726678
You put the numbers from least to greatest.\nthen you get to the number that is in the middle of the set. (it's easy to do this by crossing off the numbers on the outsides till you get to the middle)\n\nStep A) 23, 25, 26, 30, 37, 39\nStep B) 25, 26, 30, 37\nStep C) 26, 30\n in this case you have 2 middle numbers, so you average them (add them together, divide by two)\n26+30=56\n56 divided by 2 is\n28\nSo yes, 28 is your median.
1147092
28\n\nleap years (which have Feb.29-th) are all divisible by 4
342569
divide by a number and get the same number you are dividing by as an answer for example  \n49 divided by 7 =7
1098310
There are lots of them.  The easy way is to multiply 5 times 6 times 7.  This number is divisible by 5, 6, and 7.  So -- just add two to it.\n\nThis is 212.\n\nYou want another?  How about 2102?  210000000002?\n422? 420002?\n\nI got a million of 'em.
195987
although there can be some patterns seen in the numbers it is not a repeating decimal and can be taken out to a lot more decimal places then humans can calculate.  Also if you put it into fraction (22/7) that is the most accurate way to actually utilize the number.
779216
Each digit is even\nEach digit is different\nThe number has 4 digits\nIt is divisible by 4.\nIt is not divisible by 8.\nIt is a square.\n\nYour answer must be: 6084\nIt is the square of 78.\nAll digits are even and different.\nIt is divisible by 4.\nIt is not divisible by 8
913443
Test for divisible by 2 - last digit is 0, 2, 4, 6, 8\nTest for divisible by 4 - last *two* digits are divisible by 4\nTest for divisible by 5 - last digit is 0 or 5\nTest for divisible by 6 - divisible by 2 and sum of digits is a multiple of 3).\nTest for divisible by 9 - sum of digits is a multiple of 9\nTest for divisible by 10 - last digit is 0.\n\nIf you check 1800, you'll see that it passes ALL tests.
164857
1010020\n1_1_0_0 summing digits results in 2.\n_0_0_2_ summing digits results in 2.\nSo it is divisible by 11.\n\nAnother way:\n1 - 0 + 1 - 0 + 0 - 2 + 0 = 0\nSo 1010020 is divisible by 11.
446269
Just put the question from Mathmatters 2 whether it is even or odd on Yahoo! Answers and you will find the answer in different methods within seconds.
1143446
I do both of those and I also test numbers for divisibility.  My car license plate has as its numeric part the number 594 which is divisible by 11, that's how I remember my license plate number.  Also, when someone asks me to pick a number between 1 and 10, I usually pick pi, phi, or e, just to be a smart@$$.  They didn't say an INTEGER between 1 and 10, did they???
20180
You can't divide by 0, you can divide anything by 1, you can divide any even number by 2 to get a whole number,...etc. This website has more divisibility rules: \nhttp://mathforum.org/k12/mathtips/division.tips.html
8949
If you have an odd number of numbers, it's true.\nE.g., given the set:   1,   2,   3\nthe median is 2.\n\nIf you have an even number of numbers, it's false.\nE.g., given the set:  1,   2,   3,   4\nthe median is 2 1/2  (the average of 2 and 3)
87359
Check if the year is divisible by both 4 and 400. If divisible it is a sure leap year. elseif it was divisible only by 100, it is not be a leap year.
728995
7 is a prime number.  It is divisible only by itself and 1.