Hi sud21,
There is a pattern involving prime factorization that can help on this question. While it's a relatively rare pattern, the GMAT might slip in into your Test in 1 question (when you're scoring at a sufficiently high level).
For a positive integer to have EXACTLY 6 factors, the prime-factorization of that number will yield 2 DIFFERENT primes, one of which shows up TWICE.
For example:
12 = (2)(2)(3) = 2 different primes, one of which shows up twice.
Factors of 12: 1,12, 2,6, 3,4
We can use this pattern to quickly find the various integers that fit this pattern (and 'leapfrog' all of the integers that don't). Since the two Facts "hint" at the fact that we're going to be dealing with 2-digit numbers, I'm going to limit my work to those.....
Start with the prime that shows up twice:
(2^2)(3) = 12
(2^2)(5) = 20
x7 = 28
x11 = 44
x13 = 52
x17 = 68
X19 = 76
Etc.
(3^3)(2) = 18
(3^3)(5) = 45
x7 = 63
x11 = 99
(5^2)(2) = 50
(5^2)(3) = 75
(7^2)(2) = 98
1) A and (A+1) each have exactly 6 factors.
IF...
A = 44, (A+1) = 45, then the answer to the question is 44
IF...
A = 75, (A+1) = 76, then the answer to the question is 75
Fact 1 is INSUFFICIENT
2) A < 76
Here, A can be any integer from 1 to 75, inclusive.
Fact 2 is INSUFFICIENT
Combined, we have enough information in our prior work to prove that A could be 44 or 75.
Combined, INSUFFICIENT
Final Answer: E
GMAT assassins aren't born, they're made,
Rich
Number properties
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem.
Remember equal number of variables and independent equations ensures a solution.
What is the value of the positive integer a?
(1) a and (a+1) each have exactly 6 factors.
(2) a < 76
In the original condition there is 1 variable (a) and thus we need 1 equation to match the number of variable and equation. Since there is 1 each in 1) and 2), D has high probability of being the answer. In order to have 6 factors, we must have (p^2)q while p and q are distinct prime numbers.
--> number of factors = (2+1)(1+1)=6
In case of 1), a=44=(2^2)11, a+1=45=(3^2)5, a=75=3(5^2), a+1=76=(2^2)19....the answer is not unique and therefore the condition is not sufficient
In case of 2), the answer is not unique and therefore the condition is not sufficient.
Using both 1) & 2) together, a=44=(2^2)11, a+1=45=(3^2)5, a=75=3(5^2), a+1=76=(2^2)19....the answer is still not unique, and therefore the conditions are not sufficient. Therefore the answer is E.
Normally for cases where we need 1 more equation, such as original conditions with 1 variable, or 2 variables and 1 equation, or 3 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore D has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) separately. Here, there is 59 % chance that D is the answer, while A or B has 38% chance. There is 3% chance that C or E is the answer for the case. Since D is most likely to be the answer according to DS definition, we solve the question assuming D would be our answer hence using 1) and 2) separately. Obviously there may be cases where the answer is A, B, C or E.
Math Revolution : Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World's First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
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Remember equal number of variables and independent equations ensures a solution.
What is the value of the positive integer a?
(1) a and (a+1) each have exactly 6 factors.
(2) a < 76
In the original condition there is 1 variable (a) and thus we need 1 equation to match the number of variable and equation. Since there is 1 each in 1) and 2), D has high probability of being the answer. In order to have 6 factors, we must have (p^2)q while p and q are distinct prime numbers.
--> number of factors = (2+1)(1+1)=6
In case of 1), a=44=(2^2)11, a+1=45=(3^2)5, a=75=3(5^2), a+1=76=(2^2)19....the answer is not unique and therefore the condition is not sufficient
In case of 2), the answer is not unique and therefore the condition is not sufficient.
Using both 1) & 2) together, a=44=(2^2)11, a+1=45=(3^2)5, a=75=3(5^2), a+1=76=(2^2)19....the answer is still not unique, and therefore the conditions are not sufficient. Therefore the answer is E.
Normally for cases where we need 1 more equation, such as original conditions with 1 variable, or 2 variables and 1 equation, or 3 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore D has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) separately. Here, there is 59 % chance that D is the answer, while A or B has 38% chance. There is 3% chance that C or E is the answer for the case. Since D is most likely to be the answer according to DS definition, we solve the question assuming D would be our answer hence using 1) and 2) separately. Obviously there may be cases where the answer is A, B, C or E.
Math Revolution : Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World's First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
Unlimited Access to over 120 free video lessons - try it yourself (https://www.mathrevolution.com/gmat/lesson)
See our Youtube demo (https://www.youtube.com/watch?v=R_Fki3_2vO8)
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[email protected] wrote:For a positive integer to have EXACTLY 6 factors, the prime-factorization of that number will yield 2 DIFFERENT primes, one of which shows up TWICE.
Not quite right, p� would also have exactly six factors. (2�, for instance: 1, 2, 4, 8, 16, 32.)Max@Math Revolution wrote: In order to have 6 factors, we must have (p^2)q while p and q are distinct prime numbers.
