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Z=2a×5b×7c A positive integer Z can be expressed in terms of its prime

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Z=2a×5b×7c A positive integer Z can be expressed in terms of its prime [#permalink]

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New post 23 May 2017, 00:34
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\(Z=2^a×5^b×7^c\)

A positive integer Z can be expressed in terms of its prime factors as above, where a, b and c are positive integers. Is \(3^{|a−b|}<9\)?

(1) Z is divisible by 40 but not by 50
(2) \(Z^{a−b}=2^6×5^2×7^4\)
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Z=2a×5b×7c A positive integer Z can be expressed in terms of its prime [#permalink]

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New post 23 May 2017, 00:50
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\(Z=2^a×5^b×7^c\)

A positive integer Z can be expressed in terms of its prime factors as above, where a, b and c are positive integers. Is \(3^{|a−b|}<9\)?


Is \(3^{|a−b|}<9\)?

Is \(3^{|a−b|}<3^2\)?

Is \(|a−b|<2\)?


(1) Z is divisible by 40 but not by 50

\(40 = 2^3*5\) is a factor of Z while \(50 = 2*5^2\) is not. This implies that a (the power of 2) is at least 3 and b (the power of 5) is exactly 1: \(a > 3\) and \(b = 1\). Thus, \(|a−b|\) will be greater than 2. Sufficient.


(2) \(Z^{a−b}=2^6×5^2×7^4\).

\(Z^{a−b}=2^6×5^2×7^4=(2^a×5^b×7^c)^{a-b}\).

\(2^6×5^2×7^4=(2^a×5^b×7^c)^{a-b}\).

\(2^6×5^2×7^4=2^{a(a-b)}×5^{b(a-b)}×7^{c(a-b)}\).

\(6 = a(a-b)\) and \(2 = b(a-b)\)

Divide one by another \(3 = \frac{a}{b}\) --> \(a = 3b\).

\(|a−b|=|2b|\). Since b is a positive integer, then |2b| is at least 2. Thus, \(|a−b|=|2b|\) will be greater than 2. Sufficient.

Answer: D.

Hope it'clear.
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Re: Z=2a×5b×7c A positive integer Z can be expressed in terms of its prime [#permalink]

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New post 23 May 2017, 15:59
Bunuel, 40 = 2^3*5 and 50 = 2*5^2.
I agree that a is atleast 3.. but how is b EXACTLY 1.. I can do Z= 200, which is 2^3*5^2*7^0 .. so b can be 2 as well right..
am I missing something?
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Re: Z=2a×5b×7c A positive integer Z can be expressed in terms of its prime [#permalink]

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New post 23 May 2017, 21:40
sasidharrs wrote:
Bunuel, 40 = 2^3*5 and 50 = 2*5^2.
I agree that a is atleast 3.. but how is b EXACTLY 1.. I can do Z= 200, which is 2^3*5^2*7^0 .. so b can be 2 as well right..
am I missing something?


We are told that z is divisible by 5 but not by 5^2, so the power of 5 in z must be 1.
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Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Z=2a×5b×7c A positive integer Z can be expressed in terms of its prime   [#permalink] 23 May 2017, 21:40
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