SOLUTION: Suppose there is a pile of quarters, dimes, and pennies with a total of $1.07. a) How much of each coin can be present without being able to make change for a dollar? b)Explain w

ANSWER.  There are two and only two possible collections :

          - one is  { 2 pennies, 3 quarters and 3 dimes } 

          - and the other is  { 2 pennies, 1 quarter and 8 dimes }.



Explanation.  


1)  The only alternative to 2 pennies is to have 7 pennies, or 12 pennies, or 17 pennies (and so on . . . ) to make a total of $1.07.


    But these alternatives contradict to be not able to make a change for a dollar.


    So, 2 pennies is the only option, regarding pennies.



2)  OK.  It leaves us with $1.05.


    Then we must be able to make $1.05 using quarters and dimes only, without being able to make change for a dollar.


    Again, it is obvious that there are only two such opportunities : 

        - one quarter and  8 dimes,

    and/or

        - three quarters and 3 dimes.