Euler problems/71 to 80

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Problem 71

Listing reduced proper fractions in ascending order of size.


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Problem 72

How many elements would be contained in the set of reduced proper fractions for d ≤ 1,000,000?


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Problem 73

How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions?


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Problem 74

Determine the number of factorial chains that contain exactly sixty non-repeating terms.


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Problem 75

Find the number of different lengths of wire can that can form a right angle triangle in only one way.

Solution: This is only slightly harder than problem 39. The search condition is simpler but the search space is larger.

problem_75 = length . filter ((== 1) . length) $ group perims
    where  perims = sort [scale*p | p <- pTriples, scale <- [1..10^6 `div` p]]
           pTriples = [p |
                       n <- [1..1000],
                       m <- [n+1..1000],
                       even n || even m,
                       gcd n m == 1,
                       let a = m^2 - n^2,
                       let b = 2*m*n,
                       let c = m^2 + n^2,
                       let p = a + b + c,
                       p <= 10^6]

Problem 76

How many different ways can one hundred be written as a sum of at least two positive integers?


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Problem 77

What is the first value which can be written as the sum of primes in over five thousand different ways?


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Problem 78

Investigating the number of ways in which coins can be separated into piles.


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Problem 79

By analysing a user's login attempts, can you determine the secret numeric passcode?


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Problem 80

Calculating the digital sum of the decimal digits of irrational square roots.


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