2k is a perfect square, 3k is a perfect cube, and 5k is a perfect 4th power. What is the sum of the exponents in the prime factorization of the smallest such positive integer k ?
The prime factorization of k is :
k=pn11⋅pn22⋅...⋅pnmm
That factorization must contain 2, 3 and 5 as prime numbers :
⇒k=2n1⋅3n2⋅5n3
n1 must be odd, multiple of 5, and multiple of 3⇒15
n2 must be 2 modulo 3, even, and multiple of 5⇒20
n3 must be 4 modulo 5, even, and multiple of 3⇒24
So, 15+20+24=59 is the number asked.
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