The "This and That" section will be a compilation of various and sundry items which are helpful but might not necessarily fall into any specific category. The first of these is "Divisibility".
Many times (such as factoring, cancelling, removing perfect squares from radicals) one needs to know if a number is divisible by another number.
For example, if I had 34212*(2/3) could I cancel the 3.
And it would be nice if I knew this without actually going to the trouble of doing the math.
Now, merely by looking at these number, I have no clue. BUT there is a trick.
Add all the digits up of 34212 and they add up to 12 .
And 12 is evenly divisible by 3, so therefore 342012 is evenly divisible by 3.
In fact, 3 will go into 34212 a total of 11404 times. Said another way 34212/3 = 11404.
But what is really important from the above is the trick one can use to determine if 3 will go into the number an even number of times.
And that trick is "add up the digits in the number ... if 3 will go evenly into this sum then it goes into that original number evenly".
And there are other tricks. In fact, here are some of these rules.
An integer is divisible by 2 iff (if and only if) its right-most digit is one of these numbers 0,2,4,6,8 (or you could say it is divisible by 2 iff it is an even number) Example: 129976 is divisible since it ends in 6 (one of the even numbers)
An integer is divisible by 3 iff (if and only if) the sum of its digits is divisible by 3. Example: 111126 is divisible by 3 since 1+1+1+1+2+6 = 12 is divisible by 3
An integer is divisible by 4 iff (if and only if) its right-most pair of digits is divisible by 4 Example: 119824 is divisible by 4 since 24 is divisible by 4 However 321994 is not divisible by 4 since 94 is not divisible by 4
An integer is divisible by 5 iff (if and only if) its right-most digit is one of these numbers 0 or 5 Example: 432880 is divisible since it ends in 0
An integer is divisible by 6 iff (if and only if) it is divisible by 2 and 3 (see the rules for 2 & 3 above). This means that the sum of the digits must be divisible by 3 AND the number in question must end in one of the following 0,2,4,6,8 Example: 43872 is divisible since it is even (ending in a 0,2,4,6 or 8) AND the sum of the digits is divisible by 3 (4+3+8+7+2 = 24 and 24 is divisible by 3)
Sorry, no special rule for divisibility by 7.
An integer is divisible by 8 iff (if and only if) its right-most three digits is divisible by 8. Example: 103120 is divisible by 8 since 120 is divisible by 8 However 52218 is not divisible by 8 since 218 is not divisible by 8
An integer is divisible by 9 iff (if and only if) the sum of its digits is divisible by 9. Example: 51039 is divisible by 9 since 5+1+0+3+9 = 18 is divisible by 9 However 2665 is not divisible by 9 since 2+6+6+5=19 is not divisible by 9
An integer is divisible by 10 iff (if and only if) its last digit (right-most digit) is a 0 Example: 22760 is divisible by 10 However 82771 is not divisible by 10
An integer is divisible by 25 iff (if and only if) its last two digits (right-most digit) are either 00 or 25 or 50 or 75 Example: 22700 22925 23450 and 29975 are divisible by 25 However 82771 is not divisible by 25
An integer is divisible by 100 iff (if and only if) its last two digits (right-most digit) are 00 Example: 22700 is divisible by 100 However 22738 is not divisible by 100
Additional references on the subject of "divisibility":
Math Is Fun
Math Goodies
And wikipedia is always on of the best sites for such information: