If \(3^x(5^2)\) is divided by \(3^5(5^3)\), the quotient terminates with one decimal digit. If x > 0, which of the following statements must be true?

(A) x is even
(B) x is odd
(C) x < 5
(D) x ≥ 5
(E) x = 5

Note: The quotient terminates with one decimal digit. Fraction * fraction yields more decimal points. 0.2 * 0.2 = 0.04 and 2 * 0.2 = 0.4. we have 0.4 type result. thus one must be integer and the other is fraction.

Given,

\(\frac{3^x 5^2}{3^55^3}\)

= \(3^{(x-5)} 5^{(2-3)}\)

= \(3^{(x-5)} 5^{(-1)}\)

=\(3^{(x-5)}* \frac{1}{5}\)

\(\frac{1}{5} = 0.2\)

Thus , \(3^{(x-5)}\) has to an integer.

\(x\geq5\) is a must to have integer in this case.

Thus the best answer is D.

\(\frac{3^x(5^2)}{3^5(5^3)}\), this quotient terminates , implies that x is an integer.

You know 2 or 5 or both 2 and 5, in the denominator always terminates. When a multiple of 3 is divided by 3, then the quotient is never terminated.

So, as long as \(3^5\) is active in the denominator, the quotient won't terminate. We need to cancel out \(3^5\) in the denominator in order to get a terminating decimal of the quotient.

So, x=5 satisfies the condition. however, we need to find out the MUST BE TRUE condition, which is true at all points of the condition rather than at a single point.

So, \(x\geq5\)

Ans. (D)
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