Puzzles Involving Number 17

17 the 7-th prime number and the only prime of the form p^q + q^p, where p and q are prime: 17 = 2^3 + 3^2

the first prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …

17 is the only prime number that is the sum of four consecutive primes (2 + 3 + 5 + 7).

17 is the lowest number that can be written as A^3 + B^2 in 2 distinct ways:
17 = 2^3 + 3^2 and 17 = 1^3 + 4^2

17 is a Fermat prime and it is also the exponent of a Mersenne prime (a prime p for which 2^p – 1 is prime).

Puzzles :

(1) Why is a number of the form abcdefghabcdefgh is divisible by 17 (replace each letter with a digit. E.g. 1234567812345678)?

(2) The wallpaper groups are the 17 possible plane symmetry groups.

(3) The problem of the 17 camels : A sheik has 3 children and owns 17 camels. His will stipulates that the eldest is to receive half his property; the second son is to receive the third of his property; and the third one, the ninth of his property. On his death, how would the sons share out the inheritance?

(4) x and y are integers between 2 and 100. Alex knows S = x+y, and Pat knows P = xy, but they do not know x and y. “I can’t calculate them” Pat says, “I knew” Alex says. “So I know these two numbers” Pat says, “In this case, so do I” Alex concludes.

Solution?

(5) Autoreferential growth rate : an enterprise growing at the rate of exactly 17.65717% per year will become exactly 17.65717 times larger in exactly 17 years, seven months, 29 days, and 21 hours… Which is exactly 17.65717 years!

See also: Puzzles Involving Numbers 3

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