find the number of integers smaller than and are relatively prime to 80

            80
           /  \
         10    8
         /\    /\
        2  5  2  4 
                 /\
                2  2

The only prime factors of 80 are 2 and 5. So those
relatively prime to 80 cannot be divisible by 5.
So they cannot end in an even digit or 5.  So they
must end in either 1,3,7, or 9

We will think of 1-digit numbers as 2-digit numbers
with first digit 0, like 1 = 01, 2 = 02, etc.

To be less than 80 the first digit can only be 0,1,...,7
So there are 8 choices for the first digit.

For each of those 8 choices for the first digit,
there are 4 choices for the last digit 1,3,7,9,

Answer 4×8 = 32
  
Here they all are:

 1,  3,  7,  9, 11, 13, 17, 19,
21, 23, 27, 29, 31, 33, 37, 39, 
41, 43, 47, 49, 51, 53, 57, 59, 
61, 63, 67, 69, 71, 73, 77, 79.

Edwin