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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If 3^x(5^2) is divided by 3^5(5^3), the quotient terminates with one

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Math Expert V
Joined: 02 Sep 2009
Posts: 59561
If 3^x(5^2) is divided by 3^5(5^3), the quotient terminates with one  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 62% (02:01) correct 38% (01:54) wrong based on 64 sessions

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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
NUS School Moderator V
Joined: 18 Jul 2018
Posts: 1023
Location: India
Concentration: Finance, Marketing
WE: Engineering (Energy and Utilities)
Re: If 3^x(5^2) is divided by 3^5(5^3), the quotient terminates with one  [#permalink]

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$$\frac{3^x5^2}{3^55^3}$$ = $$\frac{3^x}{3^55}$$. Since $$\frac{1}{5}$$ =0.2. Inorder for a fraction to have a terminating decimal, the denominator should only consist of either 2s or 5s.

Hence $$x\geq{5}$$.
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Joined: 31 Oct 2013
Posts: 1489
Concentration: Accounting, Finance
GPA: 3.68
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If 3^x(5^2) is divided by 3^5(5^3), the quotient terminates with one  [#permalink]

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1
Bunuel wrote:
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.
Intern  B
Joined: 04 May 2014
Posts: 45
Concentration: Strategy, Operations
Re: If 3^x(5^2) is divided by 3^5(5^3), the quotient terminates with one  [#permalink]

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selim wrote:
Bunuel wrote:
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.

Thanks Selim for the explanation.
I also solved the problem but by substituting different values of x (x>0). It took a bit longer time as I couldn't think of the logic of 3 ^(x-5) to be an integer.
How to develop such logics ? Pls suggest.

BTW, my answer- fraction will be reduced to $$3^{(x-5)}* \frac{1}{5}$$
If x=1, $$1/(3)^{4}* 1/5$$, since 1/3 itself is non terminating so the whole fraction is non-terminating
If x=2, x=3, x=4, same issue as in x=1
If x=5, fraction =1/5 =0.2; so x can be 5
If x=6, fraction = 3/5 = 3 x 0.2 = 0.6, so x can be 6
If x=7, fraction = 9/5 = 9 x 0.2 = 1.8; so x can be 7
So on so forth, we can conclude from the above results that x>=5. Hence, answer is D
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If 3^x(5^2) is divided by 3^5(5^3), the quotient terminates with one  [#permalink]

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Bunuel wrote:
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

$$\frac{3^x(5^2)}{3^5(5^3)}$$, 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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PKN

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VP  D
Joined: 31 Oct 2013
Posts: 1489
Concentration: Accounting, Finance
GPA: 3.68
WE: Analyst (Accounting)
If 3^x(5^2) is divided by 3^5(5^3), the quotient terminates with one  [#permalink]

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tatz wrote:
selim wrote:
Bunuel wrote:
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.

Thanks Selim for the explanation.
I also solved the problem but by substituting different values of x (x>0). It took a bit longer time as I couldn't think of the logic of 3 ^(x-5) to be an integer.
How to develop such logics ? Pls suggest.

BTW, my answer- fraction will be reduced to $$3^{(x-5)}* \frac{1}{5}$$

[color=#ff0000]If x=1, $$1/(3)^{4}* 1/5$$, since 1/3 itself is non terminating so the whole fraction is non-terminating

If x=2, x=3, x=4, same issue as in x=1
If x=5, fraction =1/5 =0.2; so x can be 5
If x=6, fraction = 3/5 = 3 x 0.2 = 0.6, so x can be 6
If x=7, fraction = 9/5 = 9 x 0.2 = 1.8; so x can be 7
So on so forth, we can conclude from the above results that x>=5. Hence, answer is D[/color]

Bro , i applied the same concept as u described above. I did it directly. If u put any value for x less than 5 it gives us non terminating decimal but we need one terminating decimal. Thus i said that 3^{(x-5)} has to be an integer. Any value of x less than 5 will give us negative exponent but if $$x \geq5$$ we can't avoid the problem.

Thanks. If 3^x(5^2) is divided by 3^5(5^3), the quotient terminates with one   [#permalink] 14 Aug 2018, 01:54
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