Must be a factor of y (interpretation of factors)

[netcaesar]

If x and y are integers and 63^2 ×10^x = 15^4 y , what must be a factor of y?
a. 21
b. 35
c. 980
d. 784
e. 1176

As soon as I finish the prime factorizarion I do not understand what I must do because I do not have the concepts clear.

OA is D

[Guest]

Source: Gmat Guru

[sanjaylakhani]

Equation can be simplified to :

y= 7^2*5^(x-4)*2^x

x has to be more then 4, otherwise Y will nt be an integer.

980=2^2*5*7^2
784=2^4*7*2
1176=2^3*3*7^2

if we look at numbers above,980 is rejected because if x=4 then in Y, there is no 5

1176 is rejected as Y does nt have 3 as a factor

784 meets all conditions

[michael_shaunn]

Given is that 63^2 ×10^x = 15^4 y.
Lets simplify it.On simplification we get............3^4*7^2*2^x*5^x=3^4*5^4*y.
Looking carefully,we can say that in RHS of the above equation the factor 7 has not been taken into account.So y must contain two 7's as we are very clear about the number of 7 in the given equation.
Also we are very clear about the number of 3's on both the sides i.e all 4 3's have been covered up.

NOW we are left with the number of 2's and 5's.

On looking at both sides we can say that the equation contains atleast 4 5's if not more.Therefore the equation must also consist of atleast 4 2's on both the sides of the equation.
On the RHS of the above equation the 2's are covered up by y i.e. y contains atleast 4 2's.

So overall..we can say for sure that y contains 2 7's and atleast 4 2's which gives the the least possible value of y as 7^2*2^4=784.
thanks!!

[JonathanSchneider]

Agreed.