Euler problems/61 to 70

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Revision as of 01:03, 30 March 2007 by Quale (talk | contribs) ([ Problem 63]: a solution)

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Problem 61

Find the sum of the only set of six 4-digit figurate numbers with a cyclic property.


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Problem 62

Find the smallest cube for which exactly five permutations of its digits are cube.


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Problem 63

How many n-digit positive integers exist which are also an nth power?

Solution: Since dn has at least n+1 digits for any d≥10, we need only consider 1 through 9. If dn has fewer than n digits, every higher power of d will also be too small since d < 10. We will also never have n+1 digits for our nth powers. All we have to do is check dn for each d in {1,...,9}, trying n=1,2,... and stopping when dn has fewer than n digits.

problem_63 = length . concatMap (takeWhile (\(n,p) -> n == nDigits p))
             $ [powers d | d <- [1..9]]
    where powers d = [(n, d^n) | n <- [1..]]
          nDigits n = length (show n)

Problem 64

How many continued fractions for N ≤ 10000 have an odd period?


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Problem 65

Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.


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Problem 66

Investigate the Diophantine equation x2 − Dy2 = 1.


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Problem 67

Using an efficient algorithm find the maximal sum in the triangle?


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Problem 68

What is the maximum 16-digit string for a "magic" 5-gon ring?


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Problem 69

Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.


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Problem 70

Investigate values of n for which φ(n) is a permutation of n.


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