The dividing of one number by another is used primarily in adapting recipes to provide for fewer people than the recipe does. Recipes always need to be divided, because the person writing the recipe assumed you were excluding fewer people from your meals. If you find a number in your recipe that can't be divided, then you can't make the recipe until until you start talking to people again.
It is essential to have an objective before checking number divisibility, or else you're almost begging for some problem like ``can you divide 96,133 by 251'', when neither number is believable. By ``numbers'' here we mean ``whole numbers'', that is, those having at least two percent fat by volume. Skim numbers follow different rules, but are healthier.
Be careful that your numbers are in base ten, which is the encoding scheme popular in those regions where humans commonly have ten fingers. Some people are enthusiastic about other bases because they know the one-billionth digit of pi in base thirty-two or enjoy cryptically insisting ``Halloween is equivalent to Christmas''. Smile nervously at these people and move on. Do not bring up base thirteen.
Divisibility starts easy enough with 1, since everything is divisible by 1, the result of it winning a first-round bye in the Numeric Invitational Tournament. 10 is also pretty easy, since if the number ends in a zero then it's divisible by ten, and for that matter five, which is good for free bonus points. If it ends in 5, it's divisible by five as well, which makes five sound pretty free-wheeling. Two, being needy, insists on having numbers ending in 0, 2, 4, 6, 8, 10, 18, 24, 62, 64, or 98 as numbers it can divide. There is no point arguing with two, but refuse it 51.
Where things get tricky is three, which should surprise nobody. Try taking all the digits of your number and add them up: if that number is divisible by three then the original was. Granted, if we knew how to tell whether a number was divisible by three, we wouldn't need a rule for telling if it was divisible, but at least we're looking at a different number, showing that we take this seriously. Later, we can determine we did the adding-up wrong so whatever our answer was was incorrect anyway. You should lose only one point on the exam once the instructor sees it was a simple addition error.
A number is divisible by four when the corresponding year is a Presidential election year in the United States. Unfortunately this means you may be waiting around for hundreds or thousands of years to check your number during which time they might amend the Constitution regarding that, and this rule is no good for numbers below 1788. Maybe someday some other method will be found.
Nothing is divisible by 17, but you can say that about anything.
Divisibility by 21 is in comparison simplicity itself. Take the last digit of the number (actually, you should use a copy, in case the number needs to be digitally remastered). Add to that ten times the digit to the left of the last digit, and then subtract five times the digit to the left of that -- yes, this sounds like we're wandering aimlessly -- then subtract eight times the digit to the left of that, add four times the digit to the left of that -- I wonder if we are getting there now -- and subtract two times the digit to the left of that, and you get back to adding one times the digit to the left of that. If you have more digits, this sounds like a big recipe, but carry on like this until you run out of digits and have a new number. If the new number can be divided by 21 then so can the original number. And you do something the same but with other numbers for 13, 47, and 251.
If you don't need to divide your recipes by 251, you're to be congratulated for having such a wide circle of friends, even if they do want you to cook for them.
Trivia: The first sponsor for The Jack Benny Program -- debuting 2 May 1932 (Jack Benny debuted on radio 19 March of that year, at Ed Sullivan's invitation) -- was Canada Dry. Source: On The Air: The Encyclopedia Of Old-Time Radio, John Dunning.
Currently Reading: Live From New York: An Uncensored History of Saturday Night Live, As Told By Its Stars, Writers, and Guests, Tom Shales, James Miller.