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!
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.
But I disagree with them- it should be B and B. Please can you take a look thanks.
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.
Hope it's clear.