What is the smallest positive integer x such that 450x is the cube of

Hi Bunuel,

Great question. The way I'd suggest to solve this problem is to just Dive In.

We want to know the smallest x that will make 450x a CUBE of some number. Let's call that number y.

Let's first figure out what we're working with. The prime factorization of 450 can be visualized:
...........450
......../.......\
......45.......10
...../..\....../...\
...15...3...2.....5
.../..\
..5....3

So, we have 5 * 5 * 3 * 3 * 2 that can be multiplied together to get 450. Now we need to figure out what we need to make 450 * x into a cube of y (y^3=450*x).

We have two 5s, two 3s, and one 2. To arrange these numbers in identical triples (2,3,5), we need at least one more 5, one 3, and two 2's. Each of these triples will give us the value of y (2*3*5=30), which, multiplied by itself three times, gives us 450 * x.

Looking at the factors we need to complete the triples, we get 5 * 3 * 2 * 2 = 60. We know this is the smallest number possible because prime factors by definition cannot be broken down any further.

Therefore, we can go with answer choice D.

If time permits, we can do a sanity check. We calculated that y should be 2 * 3 * 5, or 30. 30 * 30 * 30 = 27000. Also, 450 * 60 = 27000.

Hope this helps!

Hi LighthousePrep,

I did the same way till y = 2*3*5=30 in order to complete the triplet as \(3^3 * 5^3 * 2^3\).
but i did not get the below part:
Looking at the factors we need to complete the triples, we get 5 * 3 * 2 * 2 = 60. We know this is the smallest number possible because prime factors by definition cannot be broken down any further.
please help.

450 = 2 x 3^2 x 5^2 now we need two 2s, one 3 and one 5 to make it perfect cube.
So x= 2^2 x 3 x 5 = 60.

Answer is C.

Answer = C = 60

\(450 = 2^1 * 3^2 * 5^2\)

To make it a cube, all should have power of 3

\(x = 2^2 * 3^1 * 5^1 = 60\)

Let's factorize 450

450 = \(2^{1}\) * \(3^{2}\) * \(5^{2}\)

Cube has all the powers of prime integers in multiple of 3, so to make 450 a cube we need to multiply it with the below number

x = \(2^{2}\) * 3 * 5 = 60

Answer is D

since 450x is a cube of a positive no.
so 450x=2*5*5*3*3=2^1*5^2*3^2*x

hence 450x to be a cube ,x must b=2^2*5*3=60

Answer is D

Above mentioned solutions are good. Let me try a GMAT focused solution to above problem.


As per the question:
Cube root of (450 multiplied by some number x) yields a positive integer.
i.e. (450 * x)^1/3

Now since, last digit of (multiplying 450 with any integer) will be zero, the options A and B are ruled out. Now we are left with options C, D, E.
It is always good to do such sanity checks as it helps eliminate wrong answers and thereby increasing the probability of right answer.

Now checking for each solution:
C. 450 * 30 or we can try 45*3 = 135 [Not even near to a cube root of any integer].
D. 450 * 60 or we can try 45*6 = 270 [Here we can see 27 is a cube root of 3. So 450*60=27000 which is a cube root of 30]
E. 450 * 120 or we can try 45*12 = 540 [Not even near to a cube root of any integer].

So correct answer is D.

Remember: Its always good to do sanity checks which will always help to get you to the right answer in GMAT.

This is my first explanation to any question. If you like this, press Kudos and do encourage.

Wishing good luck to all.

Official Solution:

What is the smallest positive integer \(x\) such that \(450x\) is the cube of a positive integer?

A. 2
B. 15
C. 30
D. 60
E. 120

The brute-force approach would be to systematically list multiples of 450 from 450 on up, test each one to see whether it is a perfect cube (the cube of a positive integer), and choose the first multiple that meets the criterion. However, this approach is very cumbersome. Even just trying the answer choices would take a long time. In fact, without insight into the nature of cubes, it is difficult to see how we can easily test whether a number is a cube, except by cubing various integers and comparing the results to the number in question.

A more efficient approach takes advantage of a key property of perfect cubes: its prime factors come in triplets. In other words, each of its prime factors occurs 3 times (or 6 times, 9 times, etc.) in the cube's prime factorization. To see why, try cubing \(6 = (2\times3)\):
\(6 \times 6 \times 6 = (2 \times 3)(2 \times 3)(2 \times 3) = (2 \times 2 \times 2)(3 \times 3 \times 3)\)

As you can see, the 2's and 3's occur in triplets. So our goal is to make the prime factors of \(450x\) occur in triplets as well.

The first step is to break up 450 into its prime factors:
\(450 = (45)(10) = (3 \times 3 \times 5)(2 \times 5) = 2 \times 3 \times 3 \times 5 \times 5\)

How many of each prime factor do we need to complete all the triplets? We are evidently missing two 2's, one 3, and one 5. Multiplying these missing factors together, we get \(2 \times 2 \times 3 \times 5 = 60\).

Answer: D.
_________________

It is very easy question.

We need cube of 450x
Let a^3=450x
Prime factors of x = 5^2*3^2* 2
To make the number cube we need three 5, three 3 and three 2
So we need 5*3*2^2 = 60

For this problem, determine the prime factorization of 450. This yields two factors of both 3 and 5 with one factor of 2. To get the cube of an integer, simply add two factors of 2 and a factor each of 3 and 5. Multiplying this out yield 60. Answer is D.

\(450 = 2^1 * 3^2 * 5^2\)

So, we are \(2^2*3*5 = 60\) short of a perfect cube...

Hence, in order that 450x is a perfect cube , x must be 60

Thus, Answer will be (D) 60
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450x = (y)^3
split 450
5x9x10xX = y^3
x = \(\frac{y^3}{5^2*3^2*2}\)
Check prime numbers each option 450x
We miss 60 for perfect cube
D

Cube of an integer that means the power of each prime number must be 3 or a multiple of 3.

450=45*10=2*3^2*5^2

Thus we need 2^2*3^1*5^1=60

We must remember that all perfect cubes break down to unique prime factors, each of which has an exponent that is a multiple of 3. So, let’s break down 450 into primes to help determine what extra prime factors we need to make 450x a perfect cube.

450 = 45 x 10 = 9 x 5 x 5 x 2 = 3 x 3 x 5 x 5 x 2 = 5^2 x 3^2 x 2^1

In order to make 450x a perfect cube, we need two more 2s, one more 3, and one more 5. Thus, the smallest perfect cube that is a multiple of 450 is 5^3 x 3^3 x 2^3. In other words, 450x = (5^3)(3^3)(2^3). Thus:

x = (5^3 * 3^3 * 2^3)/450

x = (5^3 * 3^3 * 2^3)/(5^2 * 3^2 * 2^1)

x = 5^1 * 3^1* 2^2 = 60

Answer: D
_________________

We need to find the value of x which will make 450x a perfect cube

Let's start factorizing 450x. We get
450x = \(2 * 3^2 * 5^2 * x\)
So, for 450x to be a perfect cube
x should be equal to \(2^2 * 3 * 5\)
Making 450x = \(2 * 3^2 * 5^2 * 2^2 * 3 * 5 \) = \(2^3 * 3^3 * 5^3\)

=> x = 4*3*5 = 60

So, Answer will be D
Hope it helps!
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