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If we observe the expression, it can be deduced to:

y= 〖3^2 (26)(5^4 (124))〗^2

Now let us eliminate options:

A: 6^4 can be eliminated as we have four 3s and four 2s
B: 62^2 can be eliminated as we have 124^2
C: 65^2 can be eliminated as we have 26^2 and 5^4
D: 15^4 can easily be eliminated
E: 52^4 cannot be eliminated as we do not have enough factors of 2

Hence answer is E
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Bunuel
OE is below:
If \(y=\frac{(3^5-3^2)^2}{(5^7-5^4)^{-2}}\), then y is NOT divisible by which of the following?
A. 6^4
B. 62^2
C. 65^2
D. 15^4
E. 52^4


\(y=\frac{(3^5-3^2)^2}{(5^7-5^4)^{-2}}=(3^5-3^2)^2*(5^7-5^4)^2=3^4*(3^3-1)^2*5^8*(5^3-1)^2=3^4*26^2*5^8*124^2=2^6*3^4*5^8*13^2*31^2\).

Now, if you analyze each option you'll see that only \(52^4=2^8*13^4\) is not a factor of \(y\), since the power of 13 in it is higher than the power of 13 in \(y\).

Answer: E.

Hi Brunel, How do you get 3^4(3^3-1)^2 from (3^5-3^2)^2?
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Bunuel
OE is below:
If \(y=\frac{(3^5-3^2)^2}{(5^7-5^4)^{-2}}\), then y is NOT divisible by which of the following?
A. 6^4
B. 62^2
C. 65^2
D. 15^4
E. 52^4


\(y=\frac{(3^5-3^2)^2}{(5^7-5^4)^{-2}}=(3^5-3^2)^2*(5^7-5^4)^2=3^4*(3^3-1)^2*5^8*(5^3-1)^2=3^4*26^2*5^8*124^2=2^6*3^4*5^8*13^2*31^2\).

Now, if you analyze each option you'll see that only \(52^4=2^8*13^4\) is not a factor of \(y\), since the power of 13 in it is higher than the power of 13 in \(y\).

Answer: E.

Hi Brunel, How do you get 3^4(3^3-1)^2 from (3^5-3^2)^2?

Factor out 3^2 from (3^5-3^2)^2: (3^2(3^3-1))^2=3^4(3^3-1)^2.

Hope it's clear.

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