Hey Everyone - I could use some help on a problem. Below is the question, I don't understand how to get to the answer. Could someone please help?
If n and y are positive integers and 450y = n^3, which of the following must be an integer?
1. y/(3x2^2X5)
2.y/(3^2x2x5)
3.y/(3x2x5^2)
I did the prime factorization of 450 and got 2x3x3x5x5, but don't understand why the answer is #1 only. How does n^3 come into play?
Thanks in advance!
Luke
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If n and y are positive integers and 450y = n^3...
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lukemacdougall
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Try to prove that I, II and III DON'T have to be integers.If n and y are positive integers and 450y = n³, which of the following must be an integer?
I. y/(3 x 2² x 5)
II. y/(3² x 2 x 5)
III. y/(3 x 2 x 5²)
a. None
b. I only
c. II only
d. III only
e. I, II, and III
To this end, plug in the MINIMUM POSSIBLE VALUE for y.
450y = n³ implies that 450y is the cube of an integer.
When we prime-factorize the cube of an integer, we get 3 (or a multiple of 3) of every prime factor:
8 is the cube of an integer because 8 = 2³ = 2*2*2.
27 is the cube of an integer because 27 = 3³ = 3*3*3.
Thus, when we prime-factorize 450y, we need to get AT LEAST 3 of every prime factor.
Here's the prime-factorization of 450y:
450y = 2 * 3² * 5² * y
Since 450 provides only one 2, two 3's, and two 5's, and we need at least 3 of every prime factor, the missing prime factors must be provided by y.
Thus, y must provide at at least two more 2's, one more 3, and one more 5.
Thus, the MINIMUM possible value of y = 2² * 3 * 5.
Plug y = 2² * 3 * 5 into the answer choices:
I. y/(3 x 2² x 5)
(2² * 3 * 5)/(3 x 2² x 5) = 1.
The smallest possible value of y yields an integer.
Eliminate every answer choice that does not include I.
Eliminate A, C and D.
II. y/(3² x 2 x 5)
(2² * 3 * 5)/(3² x 2² x 5) = 1/3.
Not an integer.
Eliminate every remaining answer choice that includes II.
Eliminate E.
The correct answer is B.
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450 = 2x3²x5²lukemacdougall wrote:If n and y are positive integers and 450y = n³, which of the following must be an integer?
1. y/(3x2²x5)
2. y/(3²x2x5)
3. y/(3x2x5²)
As 450y is a cube of a positive integer, all the prime factors 450y must be present in triplets, i.e. powers of all the prime factors of 450y in its prime factorization must be a multiple of 3.
As, 450 contains one 2, two 3s, and two 5s, y must contain at least two 2, one 3, and one 5.
Hence, y must be a multiple of 2²x3x5.
So, y/(2²x3x5) must be an integer.
But the others may or may not be an integers.
Hope that helps.
Anju Agarwal
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Brent@GMATPrepNow
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If n and y are positive integers and 450y=n^3, which of the following must be an integer?
I Y/(3 * 2^2 * 5)
II Y/(3^2 * 2 * 5)
III Y/(3 * 2 * 5^2)
A None
B I only
C II only
D III only
E I, II, and III
It's useful to work with the answer choices here. So, I've added them to the question.
It almost always helps to find the prime factorization in these question types where we ask whether a certain rational expression is an integer.
450y = n^3
2*3*3*5*5*y = n^3
For 2*3*3*5*5*y to be a cube, we need the number of 2's, 3's and 5's in the prime factorization to each be divisible by 3.
So, for example, 2*2*2*2*2*2*3*3*3*5*5*5 = (2*2*3*5)^3
For 2*3*3*5*5*y to be a cube, it must be the case that the prime factorization of y includes at least two additional 2's, one additional 3 and one additional 5.
So, y = 2*2*3*5*(other possible numbers)
Now check the option.
I. Must y/(3 * 2^2 * 5) be an integer?
Plug in y to get: 2*2*3*5*(other possible numbers)/(3 * 2^2 * 5)
= some integer
Since this must be an integer, we can eliminate A, C and D, which leaves us with B or E.
II. Must y/(3^2 * 2 * 5) be an integer?
Plug in y to get: 2*2*3*5*(other possible numbers)/(3^2 * 2 * 5)
= 2*(other possible numbers)/3
Not necessarily an integer
Since this need not be an integer, we can eliminate E, which leaves us with B.
NOTE: At this point we have the correct answer. But let's check III for "fun"
III. Must y/(3 * 2 * 5^2) be an integer?
Plug in y to get: 2*2*3*5*(other possible numbers)/(3 * 2 * 5^2)
= 2*(other possible numbers)/5
Not necessarily an integer
Answer: B
Cheers,
Brent
I Y/(3 * 2^2 * 5)
II Y/(3^2 * 2 * 5)
III Y/(3 * 2 * 5^2)
A None
B I only
C II only
D III only
E I, II, and III
It's useful to work with the answer choices here. So, I've added them to the question.
It almost always helps to find the prime factorization in these question types where we ask whether a certain rational expression is an integer.
450y = n^3
2*3*3*5*5*y = n^3
For 2*3*3*5*5*y to be a cube, we need the number of 2's, 3's and 5's in the prime factorization to each be divisible by 3.
So, for example, 2*2*2*2*2*2*3*3*3*5*5*5 = (2*2*3*5)^3
For 2*3*3*5*5*y to be a cube, it must be the case that the prime factorization of y includes at least two additional 2's, one additional 3 and one additional 5.
So, y = 2*2*3*5*(other possible numbers)
Now check the option.
I. Must y/(3 * 2^2 * 5) be an integer?
Plug in y to get: 2*2*3*5*(other possible numbers)/(3 * 2^2 * 5)
= some integer
Since this must be an integer, we can eliminate A, C and D, which leaves us with B or E.
II. Must y/(3^2 * 2 * 5) be an integer?
Plug in y to get: 2*2*3*5*(other possible numbers)/(3^2 * 2 * 5)
= 2*(other possible numbers)/3
Not necessarily an integer
Since this need not be an integer, we can eliminate E, which leaves us with B.
NOTE: At this point we have the correct answer. But let's check III for "fun"
III. Must y/(3 * 2 * 5^2) be an integer?
Plug in y to get: 2*2*3*5*(other possible numbers)/(3 * 2 * 5^2)
= 2*(other possible numbers)/5
Not necessarily an integer
Answer: B
Cheers,
Brent
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lukemacdougall
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