450y=n^3
3²*5²*2*y=n^3
y must at least contain 3,5, and 2² to suit the form n^3
Imagine y=1, 3²*5²*2=n^3 does not exist at least for n as an integer, we need to state that y contains integers to suit n^3
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gmat prep algebra
Source: Beat The GMAT — Problem Solving |
Given 450y= n cubed
break down 450 = 2 * 3 squared * 5 squared
To get to the cubed form for n, y must be atleast equal to 2 squared * 3 * 5
450 y
so 450y becomes (2 * 3 squared * 5 squared) * (2 squared * 3 * 5)
If we assume take y= 2 squared * 3 * 5
we can only have (I ) y/ 2 * 3 squared * 5 squared as an integer
If y = 2 squared * 3 * 5 then (II) would be 2/3 which is not an integer
If y = 2 squared * 3 * 5 then (III) would be 2/5 which is not an integer
break down 450 = 2 * 3 squared * 5 squared
To get to the cubed form for n, y must be atleast equal to 2 squared * 3 * 5
450 y
so 450y becomes (2 * 3 squared * 5 squared) * (2 squared * 3 * 5)
If we assume take y= 2 squared * 3 * 5
we can only have (I ) y/ 2 * 3 squared * 5 squared as an integer
If y = 2 squared * 3 * 5 then (II) would be 2/3 which is not an integer
If y = 2 squared * 3 * 5 then (III) would be 2/5 which is not an integer
first, rephrase the problem. if y divided by a bunch of factors is an integer, that's just a roundabout way of saying that all those factors go into y.
so, this question is really asking: which of the following sets of factors must divide into y?
(i) two 2's, one 3, one 5; (ii) two 3's, one 2, one 5; (iii) two 5's, one 2, one 3.
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 ...).
if you don't see why this is true, try factoring out some perfect cubes for yourself. for instance, 8 = 2^3 = 2x2x2 (three 2's). 1728 = 12^3 = 2x2x2x2x2x2x3x3x3 (six 2's, three 3's). etc.
note that 450 = 10x45 = 2x5x5x3x3, but 450y = 2x5x5x3x3xY is a perfect cube. this means that y must contain ALL the missing factors to round out the sets of 3 to make a perfect cube.
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.
so it's (i) only.
--
<random comment>
it's a bit unusual of the gmat writers to construct the denominators with the 2, 3, and 5 out of numerical order. normally, their question writing is very systematic, and i'd expect the 2, 3, and 5 to appear in that order.
</random comment>
so, this question is really asking: which of the following sets of factors must divide into y?
(i) two 2's, one 3, one 5; (ii) two 3's, one 2, one 5; (iii) two 5's, one 2, one 3.
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 ...).
if you don't see why this is true, try factoring out some perfect cubes for yourself. for instance, 8 = 2^3 = 2x2x2 (three 2's). 1728 = 12^3 = 2x2x2x2x2x2x3x3x3 (six 2's, three 3's). etc.
note that 450 = 10x45 = 2x5x5x3x3, but 450y = 2x5x5x3x3xY is a perfect cube. this means that y must contain ALL the missing factors to round out the sets of 3 to make a perfect cube.
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.
so it's (i) only.
--
<random comment>
it's a bit unusual of the gmat writers to construct the denominators with the 2, 3, and 5 out of numerical order. normally, their question writing is very systematic, and i'd expect the 2, 3, and 5 to appear in that order.
</random comment>
Ron has been teaching various standardized tests for 20 years.
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Pueden hacerle preguntas a Ron en castellano
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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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Learn more about ron
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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
--
Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.
Yves Saint-Laurent
--
Learn more about ron
