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Math Expert V
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If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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Question Stats: 53% (01:27) correct 47% (01:21) wrong based on 57 sessions

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If x^2y^3 = 200, what is xy ?

(1) y is an integer
(2) x is an integer

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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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1
Quote:
If x^2y^3 = 200, what is xy ?

(1) y is an integer
(2) x is an integer

$$x^2y^3 = 200$$

i.e. $$x^2*y^3 = 2^3*5^2$$

Statement 1: y is an integer

Case 1: y = 2 and x = 5
Case 2: $$y = 2*3^2$$ and $$x = 5/3^3$$
i.e. Different values of x*y

KEY: Nowhere is it given that x also is an Integer

NOT SUFFICIENT

Statement 2: x is an integer

Case 1: y = 2 and x = 5
Case 2: $$y = 2/3^2$$ and $$x = 5*3^3$$

i.e. Different values of x*y

KEY: Nowhere is it given that y also is an Integer

NOT SUFFICIENT

Combining the two statements

Case 1: y = 2 and x = +5
Case 1: y = 2 and x = -5

i.e. x*y may be either +10 or -10 hence

NOT SUFFICIENT

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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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1
Solution:

x²y³ = 200

Observe that 200 can be factorised as:

#1. 2³ * 5² where x = 5 and y = 2 ... (i)

However, we can also say that:

#2. $$200 = [2^\frac{1}{3}]³ * 10²$$, where x = $$10$$ and y = $$[2^\frac{1}{3}]$$ ...(ii)

Or

#3. $$200 = [200^\frac{1}{2}]² * 1³$$, where x = $$[200^\frac{1}{2}]$$ and y = $$1$$ ...(iii)

#4. $$200 = [50^\frac{1}{2}]² * [4^\frac{1}{3}]³$$, where x = $$[50^\frac{1}{2}]$$ and y = $$[4^\frac{1}{3}]$$ ...(iv)

Thus, there can be multiple ways of expressing 200.

Let's now look at the 2 statements:

St1. y is an integer
Both (i) and (iii) are valid - Insufficient

St2. x is an integer
Both (i) and (ii) are valid - Insufficient

Combining:
Since both x and y are integers apparently it seems that only (i) is correct.
However, the trick here is that it doesn't say x and y are positive integers.

Thus, another possible solution is:

2³ * (-5)² where x = -5 and y = 2

Thus, even after combining the statements, we have:
xy = 10 or -10

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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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If x^2y^3 = 200, what is xy?

(1) y is an integer
x^2= 200/(y^3) where y = any integer except zero (0)
Say y= 2 , x^2= 200/8 = 25
x= +/- 5
Now xy=(5)(2)=10 or
xy = (-5)(2)=-10 (Not Sufficient)

(2) x is an integer
But x can be +ve or -ve integer
y^3= 200/(x^2)
When x= 5 , y^3 =200/25 = 8
y= 2 .: xy = 10
When x= -5, y= 2 .: xy =-10
(Not sufficient)

(1+2) we still don’t know wether x= +ve or -ve ,though we know x and y are both integers as (-/+5)^2•(2)^3 =200

Hit that E

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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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1)+2) x and y are both integers and (xy)^2.y=200

If y=2, then (xy)^2=100 and (xy)=-10 or 10. x could be - 5 or 5. We don't know whether xy is - 10 or 10.
NOT SUFFICIENT

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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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Quote:
If x^2y^3 = 200, what is xy ?

(1) y is an integer
(2) x is an integer

x^2(y^3) = 200 = 2*100 = 5^2(2^3)

(1) y is an integer insufic

y=1: x^2=200…x=√200,…xy=√200
y=2: x^2=200/8=25…x=√25=5…xy=10

(2) x is an integer insufic

x=1: y^3=200…y=cube√200=2(cube√25)…xy=2(cube√25)
x=2: y^3=200/4=50…y=cube√50…xy=2(cube√50)

(1&2) insufic

x,y=integers then x^2(y^3) = 5^2(2^3) = (-5^2)(2^3)
xy=5(2)=10, or xy=-5(2)=-10

Ans (E)
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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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Given, $$x^2y^3 = 200$$
--> Many values of $$x$$ & $$y$$ are possible
--> Possible values of $$(x, y) = {(5, 2), (-5, 2), (1, 200^{1/3}), (200^{1/2}, 1), . . . . }$$

(1) $$y$$ is an integer
--> Possible values of $$(x, y) = {(5, 2), (-5, 2), (200^{1/2}, 1), . . . . }$$
--> No Definite value of $$x*y$$ --> Insufficient

(2) $$x$$ is an integer
--> Possible values of $$(x, y) = {(5, 2), (-5, 2), (1, 200^{1/3}), . . . . }$$
--> No Definite value of $$x*y$$ --> Insufficient

Combining (1) & (2),
--> Possible values of $$(x, y) = {(5, 2), (-5, 2)}$$
--> Possible values of $$x*y = 5*2$$ or $$(-5)*2$$
--> No Definite value of $$x*y$$ --> Insufficient

Option E
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If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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Given x^2*y^3 = 200 ,

X^2*y^3 = 5^2 * 2 ^ 3.

from statement 1 ,

y is integer ,

we don't know about the value of x , it can be non integer and this can has many solutions ,

for example , x = 1/200 and y =200. Not sufficient .

From statement 2 , .

if x is an integer , same as above we can have different values of y , so we can't find the values.

OPTION E

Originally posted by BhishmaNaidu99 on 24 Jan 2020, 07:33.
Last edited by BhishmaNaidu99 on 27 Jan 2020, 01:57, edited 1 time in total.
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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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If $$x^2y^3 = 200$$, what is xy ?

(Statement1): y is an integer
If y=2, then x could be 5 or —5
—> xy could be 10 or —10
Insufficient

(Statement2): x is an integer
If x= 5, then y = 2 —> xy = 10
If x =—5, then y =2 —> xy = —10
Insufficient

Taken together 1&2,
It will have the same values of xy as statement1 and statement2 do.
( 10 or —10)

Insufficient

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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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If x^2*y^3 = 200, what is xy ?

Prime factorization of 200 is 2^3*5^2

(1) y is an integer
When y = 5, x = 2, xy = 10
When y = 1 , x = sqrt(200), xy = sqrt(200)
Insufficient

(2) x is an integer
Similarly
When x = 2, y = 5, xy = 10
When x = 1 , y = sqrt(200), xy = sqrt(200)
Insufficient

(1)+(2)
Only possibility When x = 2, y = 5, xy = 10
Sufficient

C is correct
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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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If $$x^2y^3 = 200$$, what is xy ?

(1) y is an integer
$$x^2y^3 = 200$$ = $$2^3*5^2$$

Thus, y = 2 and x = -5 OR 5
xy = -10 OR 10

INSUFFICIENT.

(2) x is an integer
Thus, y = 2 and x = -5 OR 5
xy = -10 OR 10

INSUFFICIENT.

Together 1 and 2
Thus, y = 2 and x = -5 OR 5
xy = -10 OR 10

INSUFFICIENT.

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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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Ans: E

a)y=int
if y=1, we can get one value for x..not sufficient
b)same as a..not sufficient

combined..not sufficient

x^2y^3=200=5^2*2^3=(-5)^2*2^3
not sifficient
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If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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not know whether x y are integers such that x^2y^3 = 200
#1
y is an integer
not know if x is an integer as well ; insufficient as x = fraction or root value
#2
x is an integer
not know if y is an integer as well ; insufficient as y = fraction or root value
insufficient
from 1 & 2
if both x & y are integers then only possiblity ; 5^2 * 2^3 ; = 200
or value of x & y can be fraction as well insufficient
IMO e

If x^2y^3 = 200, what is xy ?

(1) y is an integer
(2) x is an integer

Originally posted by Archit3110 on 25 Jan 2020, 09:04.
Last edited by Archit3110 on 26 Jan 2020, 23:29, edited 1 time in total.
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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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Given that x^2y^3 = 200, we are to determine the value of xy.
It is worth noting that x and y are not restricted in any way. So, x and y can be integers, rational or irrational numbers.

Statement 1: y is an integer.
Clearly insufficient because y can be 2, and y^3=8, implying x=5. Hence xy=10
However, when y=3, then y^3=27, implying x=√(200/27). xy=3*√(200/27) = 30*√(2/27) ≠ 10.

Statement 2: x is an integer.
Clearly insufficient because x can be 5, implying y is 2. xy=10
However when x=2, then y=(50)^(1/3). xy=50*2^(1/3) ≠ 10

1+2
By combining both statements, we can now conclude that x=5 and y=2, and that xy=10.

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Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer  [#permalink]

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(1) Not suff to solve
(2) Not suff to solve
Combine (1) & (2) There are countless pairs of solutions of xy.
=> Not suff

Choice E. Re: If x^2y^3 = 200, what is xy ? (1) y is an integer (2) x is an integer   [#permalink] 26 Jan 2020, 09:35
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