#2k# is a perfect square, #3k# is a perfect cube, and #5k# is a perfect 5th power. What is the sum of the exponents in the prime factorization of the smallest such positive integer #k# ?

1 Answer
MattyMatty
Dec 20, 2017

#59#

Explanation:

#"The prime factorization of k is : "#
#k = p_1^{n_1} * p_2^{n_2} * ... * p_m^{n_m}#
#"That factorization must contain 2, 3 and 5 as prime numbers :"#
#=> k = 2^{n_1} * 3^{n_2} * 5^{n_3}#
#n_1 " must be odd, multiple of 5, and multiple of 3" => 15#
#n_2 " must be 2 modulo 3, even, and multiple of 5" => 20#
#n_3 " must be 4 modulo 5, even, and multiple of 3" => 24#
#"So, " 15+20+24 = 59 " is the number asked."#

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