If $x$ and $y$ are integers and $12x + 5y^2 = 0,$ which of the following must be true?

BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

This topic has expert replies

Post new topic Post Reply

Previous Topic Next Topic

VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 am; Followed by:3 members

If $x$ and $y$ are integers and $12x + 5y^2 = 0,$ which of the following must be true?

by VJesus12 » Thu Jun 04, 2020 7:45 am

Timer

00:00

Your Answer

Global Stats

If $x$ and $y$ are integers and $12x + 5y^2 = 0,$ which of the following must be true?

(A) $x$ is a multiple of 10

(B) $x$ is a multiple of 15

(C) $x$ is a multiple of 25

(D) $x$ is not a multiple of 7

(E) $x$ is negative

[spoiler]OA=B[/spoiler]

Source: Veritas Prep

Quote

Source: Beat The GMAT — Problem Solving | Thu Jun 04, 2020 7:45 am

Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 am; Location: Grand Central / New York; Thanked: 470 times; Followed by:34 members

Re: If $x$ and $y$ are integers and $12x + 5y^2 = 0,$ which of the following must be true?

by Jay@ManhattanReview » Thu Jun 04, 2020 8:29 pm

VJesus12 wrote: ↑
Thu Jun 04, 2020 7:45 am
If $x$ and $y$ are integers and $12x + 5y^2 = 0,$ which of the following must be true?

(A) $x$ is a multiple of 10

(B) $x$ is a multiple of 15

(C) $x$ is a multiple of 25

(D) $x$ is not a multiple of 7

(E) $x$ is negative

[spoiler]OA=B[/spoiler]

Source: Veritas Prep

Given that $12x + 5y^2 = 0,$ we have $x = \dfrac{-5y^2}{12}$

Since $x$ is an integer and 5 and 12 are co-prime to each other, $y^2$ must be a multiple of 12.

Say $y^2 = 12p$, where $p$ is an integer

Again, we have $y^2 = 12p$, thus, $y = 2√{3p}$. Since $y$ is an integer, $√{3p}$ must be an integer.

Thus, $y^2=12*3p=36p$

Again, from $x = \dfrac{-5y^2}{12}=\dfrac{-5*36p}{12}= -15p$

=> $x$ must be a multiple of 15.

The correct answer: B

Hope this helps!

-Jay
_________________
Manhattan Review GMAT Prep

Locations: GMAT Classes Oakland | GMAT Tutoring | GRE Prep Courses Albuquerque | El Paso Prep Classes SAT | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.

Quote

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pm; Followed by:5 members

Re: If $x$ and $y$ are integers and $12x + 5y^2 = 0,$ which of the following must be true?

by deloitte247 » Sat Jun 06, 2020 2:07 am

$$12x+5y^2=0$$
Express the equation in terms of x
Note: x and y are integers of whole numbers
$$\frac{12x}{12}=\frac{-5y^2}{12}$$
$$x=\frac{-5y^2}{12}$$
$$If\ \ y\ =\ 1$$
$$then\ x\ =\frac{-5\left(1\right)^2}{12}=\frac{-5}{12}=-0.42$$
$$x\ is\ not\ an\ integer\ so\ y\ne1$$
$$If\ \ y\ =\ 2$$
$$then\ x\ =\frac{-5\left(2\right)^2}{12}=\frac{-20}{12}=-1.67$$
$$x\ is\ not\ an\ integer\ so\ y\ne2$$
$$If\ \ y\ =\ 3$$
$$then\ x\ =\frac{-5\left(3\right)^2}{12}=\frac{-45}{12}=-3.75$$
$$x\ is\ not\ an\ integer\ so\ y\ne3$$
$$If\ \ y\ =\ 4$$
$$then\ x\ =\frac{-5\left(4\right)^2}{12}=\frac{-80}{12}=-6.62$$
$$x\ is\ not\ an\ integer\ so\ y\ne4$$
$$If\ \ y\ =\ 5$$
$$then\ x\ =\frac{-5\left(5\right)^2}{12}=\frac{-125}{12}=-10.42$$
$$x\ is\ not\ an\ integer\ so\ y\ne5$$
$$If\ \ y\ =\ 6$$
$$then\ x=\frac{-5\left(6\right)^2}{12}=\frac{-180}{12}=-15$$
$$x\ is\ an\ integer\ so\ y=6\ or\ multiple\ of\ 6$$
$$and\ x=-15\ or\ multiple\ of\ 15$$

Answer =B

Quote

Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

Re: If $x$ and $y$ are integers and $12x + 5y^2 = 0,$ which of the following must be true?

by Scott@TargetTestPrep » Mon Jun 08, 2020 1:01 pm

VJesus12 wrote: ↑
Thu Jun 04, 2020 7:45 am
If $x$ and $y$ are integers and $12x + 5y^2 = 0,$ which of the following must be true?

(A) $x$ is a multiple of 10

(B) $x$ is a multiple of 15

(C) $x$ is a multiple of 25

(D) $x$ is not a multiple of 7

(E) $x$ is negative

[spoiler]OA=B[/spoiler]

Solution:

Let’s write:

x = 5y^2/12

So, x is a multiple of 5. Further, notice that 12 has a factor of 3 and 5 is not divisible by 3; thus y^2 must be a multiple of 3. Finally, notice that every prime factor in the prime decomposition of y^2 appears an even number of times; thus y^2 has at least two factors of 3. After one of the factors of 3 cancels with the factor of 3 in the denominator, we still have at least one more factor of 3. Thus, y^2/12 has a factor of 3, which implies that x has a factor of 3.

Since x has both a factor of 5 and a factor of 3, x must be a multiple of 15.

Answer: B

Scott Woodbury-Stewart
Founder and CEO
[email protected]

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

Quote

Post new topic Post Reply

• Page 1 of 1

If \(x\) and \(y\) are integers and \(12x + 5y^2 = 0,\) which of the following must be true?

This topic has expert replies

If \(x\) and \(y\) are integers and \(12x + 5y^2 = 0,\) which of the following must be true?

Timer

Your Answer

Global Stats

Re: If \(x\) and \(y\) are integers and \(12x + 5y^2 = 0,\) which of the following must be true?

Re: If \(x\) and \(y\) are integers and \(12x + 5y^2 = 0,\) which of the following must be true?

Re: If \(x\) and \(y\) are integers and \(12x + 5y^2 = 0,\) which of the following must be true?