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Math Expert V
Joined: 02 Sep 2009
Posts: 64174
If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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5 00:00

Difficulty:   65% (hard)

Question Stats: 58% (02:12) correct 42% (01:57) wrong based on 155 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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Manager  S
Joined: 18 Oct 2016
Posts: 128
Location: India
WE: Engineering (Energy and Utilities)
Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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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.
Director  G
Joined: 02 Sep 2016
Posts: 625
Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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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.
Target Test Prep Representative G
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2800
Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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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.

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Manager  B
Joined: 02 Feb 2016
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GMAT 1: 690 Q43 V41
Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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1
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?
Senior Manager  G
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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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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..
CrackVerbal Quant Expert P
Joined: 12 Apr 2019
Posts: 590
Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must  [#permalink]

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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?

Hello Mastermind,

Since all answer options are in terms of a and b, we try to express a in terms of b or vice versa. This approach is fairly common when an equation is given where there are two variables and the co-efficients cannot be simplified.

You see that the co-efficient of a = $$3^5$$ i.e. 243 and the co-efficient of b = $$5^3$$ i.e. 125. Clearly, 243 and 125 do not divide each other. Hence, we should think of expressing a in terms of b or vice versa.

Expressing a in terms of b, we have a = $$\frac{125b }{ 243}$$. Since 125 is not divisible by 243, b HAS to be divisible by 243. If b is divisible by 243, it will also be divisible by ALL FACTORS of 243. Therefore, $$(\frac{b}{27}$$) is definitely an integer since 27 is a factor of 243.

_________________ Re: If a and b are integers, and 3^5*a=5^3*b, which of the following must   [#permalink] 31 Mar 2020, 00:13

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