This question can be solved in the following way (I used cents instead of dollars):
Start with 100 1 cents. Then replace the cents with nickel, dime and quarter.
5 cents = 1 nickel; and the total count of coins will be reduced by 4. 10 cents = 1 dime; and the total count of coins will be reduced by 9. 25 cents = a quarter; and the total count of coins will to reduced by 24.
50 and 100 cents won't be included here because both would reduce the total count below 52.
So the ONLY ways to get an ODD number of total coins is to have 1, 3, 5 dimes, and reduce the count by 9, 27, 45. Then coupled with reduction of the count by 4, 8, 12, 16 ...(nickels) or by 24 (quarter), you can figure out the ALLOWED counts of coins.