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Q1 If 3^a*4^b = c, what is the value of b? (1) 5^a = 25 (2)

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Q1 If 3^a*4^b = c, what is the value of b? (1) 5^a = 25 (2) [#permalink] New post 10 Dec 2010, 00:00
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Q1 If 3^a*4^b = c, what is the value of b?

(1) 5^a = 25

(2) c = 36


Q2 How many distinct positive factors does the positive integer x have?

(1) x is the product of 3 distinct prime numbers.

(2) x and 3^7 have the same number of positive factors.

[Reveal] Spoiler:
OA :
Q1 C
Q2 D
But I disagree with them- it should be B and B. Please can you take a look thanks.
- Nishant

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Re: Disagree with OA [#permalink] New post 10 Dec 2010, 00:33
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nusmavrik wrote:
Q1 If 3^a*4^b = c, what is the value of b?

(1) 5^a = 25

(2) c = 36


Q2 How many distinct positive factors does the positive integer x have?

(1) x is the product of 3 distinct prime numbers.

(2) x and 3^7 have the same number of positive factors.

[Reveal] Spoiler:
OA :
Q1 C
Q2 D
But I disagree with them- it should be B and B. Please can you take a look thanks.
- Nishant


Both OA's are correct.

If 3^a*4^b = c, what is the value of b?

Note that we are not told that the variables are integers only.

(1) 5^a = 25 --> \(a=2\), but we can not get the values of \(b\). Not sufficient.

(2) c = 36 --> \(3^a*4^b = c\): it's tempting to write \(3^2*4^1=36\) and say that \(b=1\) but again we are not told that the variables are integers only. So, for example it can be that \(3^a=36\) for some non-integer \(a\) and \(b=0\), making \(4^b\) equal to 1 --> \(3^a*4^b =36*1=36\). Not sufficient.

(1)+(2) As \(a=2\) and \(c = 36\) then \(9*4^b=36\) --> \(b=1\). Sufficient.

Answer: C.


2. How many distinct positive factors does the positive integer x have?

MUST KNOW FOR GMAT:

Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.
For more on number properties check: math-number-theory-88376.html

(1) x is the product of 3 distinct prime numbers --> \(x=abc\), where \(a\), \(b\) and \(c\) are the different prime factors, so # of positive factors of \(x\) is \((1+1)(1+1)(1+1)=8\). Sufficient.

(2) x and 3^7 have the same number of positive factors --> we can get the # of factors of 3^7 (which is simply 7+1=8) and thus we know the # of factors of \(x\). Sufficient.

Answer: D.

Hope it's clear.
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Re: Disagree with OA [#permalink] New post 10 Dec 2010, 01:00
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Excellent clarification bunuel!

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Re: Disagree with OA [#permalink] New post 10 Dec 2010, 01:34
Bunuel
Pls correct my reasoning.

for Q1) my reasoning was since 3^a*4^b = c and base 3 is "even" and base 4 is "odd" then there wont be any other answer except a=2 and b=1. But I was wondering what are the other values of a and b can be for this equation to be true. Sorry, I am asking too much ;-)

for Q2) I thought that x=2^a*3^b*5^c. isnt's now x is a product of three distinct prime factors. But I relaize that I misunderstood the stem - "x is a product of distinct prime numbers". So I agree with your explanation.

Can you clarify Q1) please.

Thanks for the awesome explanation. You rock Bunuel!


Bunuel wrote:
nusmavrik wrote:
Q1 If 3^a*4^b = c, what is the value of b?

(1) 5^a = 25

(2) c = 36


Q2 How many distinct positive factors does the positive integer x have?

(1) x is the product of 3 distinct prime numbers.

(2) x and 3^7 have the same number of positive factors.

[Reveal] Spoiler:
OA :
Q1 C
Q2 D
But I disagree with them- it should be B and B. Please can you take a look thanks.
- Nishant


Both OA's are correct.

If 3^a*4^b = c, what is the value of b?

Note that we are not told that the variables are integers only.

(1) 5^a = 25 --> \(a=2\), but we can not get the values of \(b\). Not sufficient.

(2) c = 36 --> \(3^a*4^b = c\): it's tempting to write \(3^2*4^1=36\) and say that \(b=1\) but again we are not told that the variables are integers only. So, for example it can be that \(3^a=36\) for some non-integer \(a\) and \(b=0\), making \(4^b\) equal to 1 --> \(3^a*4^b =36*1=36\). Not sufficient.

(1)+(2) As \(a=2\) and \(c = 36\) then \(9*4^b=36\) --> \(b=1\). Sufficient.

Answer: C.


2. How many distinct positive factors does the positive integer x have?

MUST KNOW FOR GMAT:

Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.
For more on number properties check: math-number-theory-88376.html

(1) x is the product of 3 distinct prime numbers --> \(x=abc\), where \(a\), \(b\) and \(c\) are the different prime factors, so # of positive factors of \(x\) is \((1+1)(1+1)(1+1)=8\). Sufficient.

(2) x and 3^7 have the same number of positive factors --> we can get the # of factors of 3^7 (which is simply 7+1=8) and thus we know the # of factors of \(x\). Sufficient.

Answer: D.

Hope it's clear.

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Re: Disagree with OA [#permalink] New post 10 Dec 2010, 01:53
Expert's post
nusmavrik wrote:
Bunuel
Pls correct my reasoning.

for Q1) my reasoning was since 3^a*4^b = c and base 3 is "even" and base 4 is "odd" then there wont be any other answer except a=2 and b=1. But I was wondering what are the other values of a and b can be for this equation to be true. Sorry, I am asking too much ;-)

for Q2) I thought that x=2^a*3^b*5^c. isnt's now x is a product of three distinct prime factors. But I relaize that I misunderstood the stem - "x is a product of distinct prime numbers". So I agree with your explanation.

Can you clarify Q1) please.

Thanks for the awesome explanation. You rock Bunuel!


For 1: equation \(3^a*4^b = 36\) (if there is no restrictions on a and b) has infinitely many solutions for a and b: a=0 --> 4^b=36 --> b=log_4(36); a=1 --> 4^b=12 --> b=log_4(12), ...

For 2: x=2^a*3^b*5^c HAS 3 distinct prime factors but it's not the PRODUCT of 3 distinct prime factors, x=2*3*5 is a product of 3 distinct prime factors.

Hope it's clear.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis ; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) ; 12. Tricky questions from previous years.

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: Disagree with OA [#permalink] New post 10 Dec 2010, 02:18
Thanks Bunuel. Makes sense, the power of logarithms! I was wondering whether the first question was 700+. This is a new perspective. I havent really used logs in any of questions. Awesome !

Bunuel wrote:
nusmavrik wrote:
Bunuel
Pls correct my reasoning.

for Q1) my reasoning was since 3^a*4^b = c and base 3 is "even" and base 4 is "odd" then there wont be any other answer except a=2 and b=1. But I was wondering what are the other values of a and b can be for this equation to be true. Sorry, I am asking too much ;-)

for Q2) I thought that x=2^a*3^b*5^c. isnt's now x is a product of three distinct prime factors. But I relaize that I misunderstood the stem - "x is a product of distinct prime numbers". So I agree with your explanation.

Can you clarify Q1) please.

Thanks for the awesome explanation. You rock Bunuel!


For 1: equation \(3^a*4^b = 36\) (if there is no restrictions on a and b) has infinitely many solutions for a and b: a=0 --> 4^b=36 --> b=log_4(36); a=1 --> 4^b=12 --> b=log_4(12), ...

For 2: x=2^a*3^b*5^c HAS 3 distinct prime factors but it's not the PRODUCT of 3 distinct prime factors, x=2*3*5 is a product of 3 distinct prime factors.

Hope it's clear.

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Re: Disagree with OA [#permalink] New post 10 Dec 2010, 02:27
Expert's post
nusmavrik wrote:
Thanks Bunuel. Makes sense, the power of logarithms! I was wondering whether the first question was 700+. This is a new perspective. I havent really used logs in any of questions. Awesome !


Actually you don't really need to use logarithms (this concept is not tested on GMAT), you should just realize that since you are not told that the variables are integers only then for example if \(3^a=3^0=1\) then there will exist some non-integer \(b\) for which \(4^b =36\) thus you can not get single numerical value of \(b\) from statement (2).
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis ; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) ; 12. Tricky questions from previous years.

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: Disagree with OA [#permalink] New post 10 Dec 2010, 04:12
Very enlightening ! Thanks
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Re: Disagree with OA [#permalink] New post 10 Dec 2010, 06:06
Thanks for the explanation!
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Re: Q1 If 3^a*4^b = c, what is the value of b? (1) 5^a = 25 (2) [#permalink] New post 20 Aug 2014, 18:43
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Re: Q1 If 3^a*4^b = c, what is the value of b? (1) 5^a = 25 (2)   [#permalink] 20 Aug 2014, 18:43
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