first thing you have to do is realize the significance of the statement "450y = n^3". in this statement, the number n is actually not significant at all; this is just a roundabout way of stating that 450y is a perfect cube.
for the product 450y to be a perfect cube, its prime factors must come in threes. in other words, because the perfect cube is generated by multiplying together 3 identical copies of some number, you know that every prime factor occurs 3 times (or 6 times, or 9 times, or ...).
break down the number 450 into primes: 450 = 2 x 3 x 3 x 5 x 5.
since 450y is a perfect cube, the rest of the factors required to make a perfect cube - i.e., to complete the sets of three into which the prime factors must fall - must be part of the prime factorization of y.
so, because there's only one "2" in 450, there must be at least two more "2"s in y, in order to round out the perfect cube. (there could be more - there could be 5, or 8, or ..., but there must be at least 2.)
similar logic leads to the conclusion that there must be at least one "3" and at least one "5" in the factorization of y.
therefore, the prime factorization of y contains, at a minimum, two 2's, one 3, and one 5.
therefore (i) only. (ii) doesn't have to work because y doesn't necessarily contain two 3's, and (iii) doesn't have to work because y doesn't necessarily contain two 5's.
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Ron has been teaching various standardized tests for 20 years.
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Pueden hacerle preguntas a Ron en castellano
Potete chiedere domande a Ron in italiano
On peut poser des questions à Ron en français
Voit esittää kysymyksiä Ron:lle myös suomeksi
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Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.
Yves Saint-Laurent
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