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# If a and b are integers, and 3^5*a=5^3*b, which of the following must

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Joined: 02 Sep 2009
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If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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12 Apr 2017, 00:11
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58% (02:14) correct 42% (01:59) wrong based on 151 sessions

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If a and b are integers, and $$3^5*a=5^3*b$$, which of the following must be true?

A) b/(125) is an integer
B) a/(125*3^5) is an integer
C) b/(27) is an integer
D) a/(3) is an integer
E) a/(250) is an integer

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Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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12 Apr 2017, 03:37
Option C

: 3^5*a = 5^3*b
i.e., a = 5^3*b/3^5 & b = 3^5*a/5^3. Given a & b are integers.

b/27 = 3^2*a/5^3 = Integer always. For rest of the options, we need to ascertain the value of a & b.
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Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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12 Apr 2017, 21:48
As we have no clue about the values of a and b. And we just know that a and b are integers, options A,B,D, and E cannot be determined.

Only option C is possible here because:

3^5*a= 5^3*b
a= 5^3*b/3^5

The right hand side must be an integer as a is an integer. Thus b carries at least 3^5.

Bunuel when the OA is released, please let me know if this method is correct.
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Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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19 Apr 2017, 14:58
Bunuel wrote:
If a and b are integers, and 3^5*a=5^3*b, which of the following must be true?

A) b/(125) is an integer
B) a/(125*3^5) is an integer
C) b/(27) is an integer
D) a/(3) is an integer
E) a/(250) is an integer

We can simplify the given equation and we have:

3^5*a=5^3*b

(3^5)/(5^3) = b/a

Since (3^5)/(5^3) can’t be reduced further, we see that b is a multiple of 3^5 and a is a multiple of 5^3.

Therefore, of our answer choices, the only one that MUST be true is b/27 is an integer.

Answer: C
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Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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11 May 2017, 11:51
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How are we even simplifying the main equation given in the text? How to bring power down and put it on one side of the equation?
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Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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14 May 2017, 21:14
TheMastermind wrote:
How are we even simplifying the main equation given in the text? How to bring power down and put it on one side of the equation?

I don't the question is quoted properly. What I understood after first reading is

$$3^[5*a]=5^[3*b],$$

But the question says:

$$3^5*a = 5^3*b$$

Since we know that there are no primes common in $$3^5$$and $$5^3$$

a must contain $$5^3$$ and b must contain $$3^5$$ for inequality to hold true.

Thus choice C..
Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must &nbs [#permalink] 14 May 2017, 21:14
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# If a and b are integers, and 3^5*a=5^3*b, which of the following must

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