First part
To do this part of the problem, I simply formulated an expression and used a calculator to find out the answer. The equation was:
((((((60x3/4)-1)x7/11 -1)x5/9 -1)x2/7 - 1)x2/3-1)
The answer you get is one.
Hence, the answer is one.
Second Part
For this problem, you had to use trial and error in order to find the correct combination of fractions. After a couple of unsuccessful attempts, the answer I found was:
11/15
5/6
3/4
1/2
3/5
1/4
Third part, second question
To find the longest I can make peaches last, I took the largest integer value below 100 (99). I then started dividing it by the number's largest factor and made sure that the number when divided by the fraction, did not give us a prime number when subtracted by one.
For example, for 99:
the number I chose to divide 99 with was 32/33. This does not give us a prime number when the resultant number is subtracted by one.
We also choose the largest factor as our goal is to make the peaches last as long as they can.
The answer you get is 14 days, using the following combination of fractions:
32/33
17/19
41/42
7/9
29/31
17/19
23/25
8/9
12/13
5/7
7/8
4/5
3/5
1/2
The remainder you get is 3, and as 3 is a prime number, it can not be divided further, as per the rules given.
Hence the answer is 14 days.