If x, y, and z are integers greater than 1, and (3^27)(35^10)(z) =

(3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y)
re-written as
(3^27)(5^10)(7^10)(z) = (5^8)(7^10)(3^28)(x^y)
simplified to
25z=3(x^y) x^y is an integer, multiple of 25 so z is a multiple of 3

I z prime, so 3; x=5
II x is prime, x^y multiple of 25 so x can only be 5

D

ok so i understand up until 25 * z = 3 * (x^y)

now how do we know that z has to be a multiple of 3. is it because we know that 3 is a factor of 25 * z, and since 3 is not a factor of 25 then it has to be a factor of z.

is there such a rule like that, that if x is a factor of a*b, then x has to be a factor of one of a or b.

thx

If x, y, and z are integers greater than 1, and (3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y), then what is the value of x?

(1) z is prime
(2) x is prime


OE:


Hi, can anyone explain how this works, please.

Similar question to practice: if-x-y-and-z-are-integers-greater-than-1-and-90644.html

Hope it helps.

Aren't there 2 possible answers for #1?
x=5, y=2
x=25, y=1

Thanks!

Notice that we are told that x, y, and z are integers greater than 1, hence x=25 and y=1 is not possible.

Check here for a complete solution: if-x-y-and-z-are-integers-greater-than-1-and-57122.html#p1346892

Similar question to practice: if-x-y-and-z-are-integers-greater-than-1-and-90644.html

Hope it helps.

Oh! Can't believe I missed that! It makes sense now . Thanks!

Can anyone explain whether my approach is valid?
5^2*z = 3*x^y
(x^y)/z = (5^2)/3 = (5^2a)/(3a)
x^y = 5^2a
z = 3a

(1) z is prime, so a = 1 and x^y = 25 => x = 5
S

(2) x is prime, so a = 1 and z = 3
S

D

I just wanted to point out that \(x^y = 5^2\) is not necessarily true. In fact, if the question asked for the value of y, then statement 2 would have been insufficient

\((3^{27})(35^{10})*(z) = (5^8)(7^{10})(9^{14})(x^y)\) can be rewritten as \(\frac{(5^2 *z)}{3}=x^y\)

Written in this form, it is easy to notice that z must be a multiple of 3 since \(x^y\) is an integer. Since it is given that x is prime, the prime factorization of \(x^y\) will be x repeated y times, which means z should have at most one 3. Statement 2 doesn't require z to be prime, it could have a prime factorization of one 3 with any number of 5's and x must be 5, but y could take many integer values besides 2.

For example:

z=3*\(5^2\), x=5, y=4
z=3*\(5^3\), x=5, y=5
z=3*\(5^4\), x=5, y=6

x must be 5, so statement 2 is sufficient.

I found an almost identical question with the same question stem, but different statements.
Here it is: if-x-y-and-z-are-integers-greater-than-1-and-90644.html

There is one thing I don't understand about this problem and would appreciate any help.

When we simplify the equation and get 5^2 * (z) = 3 * (x^y), why do we even need any of the statements in the first place? Why isn't it directly clear that z is 3, x is 5 and y is 2? I really don't get it.

Thank you so much in advance.

Jay

z can take any value in that case. Think of a case in which z = 12.

\(5^2 * 3 * 2^2 = 3 * x^y\)

Here x = 10

The best way to answer this question is to use the exponential rules to simplify the question stem, then analyze each statement based on the simplified equation.

(3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y) Break up the 35^10 and simplify the 9^14
(3^27)(5^10)(7^10)(z) = (5^8)(7^10)(3^28)(x^y) Divide both sides by common terms 5^8, 7610, 3^27
(5^2)(z) = 3^xy

(1) SUFFICIENT: Analyzing the simplified equation above, we can conclude that z must have a factor of 3 to balance the 3 on the right side of the equation. Statement (1) says that z is prime, so z cannot have another factor besides the 3. Therefore z = 3.
Since z = 3, the left side of the equation is 75, so x^y = 25. The only integers greater than 1 that satisfy this equation are x = 5 and y = 2, so statement (1) is sufficient. Put differently, the expression x^y must provide the two fives that we have on the left side of the equation. The only way to get two fives if x and y are integers greater than 1 is if x = 5 and y = 2.

(2) SUFFICIENT: Analyzing the simplified equation above, we can conclude that x must have a factor of 5 to balance out the 5^2 on the left side. Since statement (2) says that x is prime, x cannot have any other factors, so x = 5. Therefore statement (2) is sufficient.

The correct answer is D.