Dividing by 9 into a mixed number

1.  First, add all the digits together and divide by 9, keeping in mind the whole number and the remainder.

2.  Write the remainder over 9, this is the fraction part of the answer.  (Make sure the fraction is in simplest form.)

3.  Add all the digits together again, except for the last one.  If there was a whole number from step 1, add this too.  Write the number down, carrying if need be.

4.  Continue to add all the digits together, subtracting off one digit each time, until you are left with the first digit. Write these numbers down, carry if need be.

5.   Write the last number down.  (Remember to add any carried numbers.)

  

Ex   8380 ÷ 9 = __________(mixed number).

a.  Add all the digits together and divide by 9: 8+3+8+0=19. 19 divided by 9 is 2 with 1 remainder.

b.  Write ¹/₉, this is the fraction of the answer.

c.  Add 8+3+8+2 = 21.  Write 1. (remember the 2)

d.  Add 8+3+2 = 13.  Write 3. (remember the 1)

e.  The last number is 8, 8+1=9 so write 9.

f.  The answer is 931 ¹/₉.

Adding a sequence (1+2+3...+n)

1+2+3+4+...+10
An easy was to do this is (n*(n+1))/2 where n is the last number (so 10 in the above example)
so 1+2+3+4+...+10=55 or (10*11)/2
An easy trick (inside the trick) is that one of the multiplied numbers will be even...
Ex. 1+2+3+...+49 is the same as 49*25 or 49*(50/2)

Happy Numbers

A happy number is as follows. Starting with any positive integer, replace that number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or until you determine that it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers.

The first 20 happy numbers are:

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100