Is positive integer A greater than positive integer B?

Hi All,

chetan2u's approach to TEST VALUES is perfect for this type of question. I do want to point out that you can do the same thing while keeping your TESTs simple.

We're told that A and B are POSITIVE INTEGERS. We're asked if A > B. This is a YES/NO question.

Fact 1: A has more factors than B.

IF...
A = 4
B = 1
A has 3 factors (1,2,4) and B has 1 factor.
The answer to the question is YES.

IF...
A = 4
B = 5
A has 3 factors (1,2,4) and B has 2 factors (1,5)
The answer to the question is NO.
Fact 1 is INSUFFICIENT

Fact 2: EVERY prime factor of B is a factor of A.

The key to dealing with this type of information is to consider that a prime factor may appear MORE than once...

IF...
A = 4
B = 2
The prime factors of B (2 only) are in A (2x2).
The answer to the question is YES.

IF....
A = 2
B = 4
The prime factors of B (2 only) are in A (2).
The answer to the question is NO.
Fact 2 is INSUFFICIENT.

Combined, we know....
A has more factors than B
EVERY prime factor of B is a factor of A

IF....
A = 4
B = 2
A has 3 factors (1,2,4) and B has 2 factors (1,2)
The prime factors of B (2 only) are in A (2x2)
The answer to the question is YES.

IF....
A = 12
B = 16
A has 6 factors (1,2,3,4,6,12) and B has 5 factors (1,2,4,8,16)
The prime factors of B (2 only) are in A (2x2x3)
The answer to the question is NO.
Combined, INSUFFICIENT.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

VERITAS PREP OFFICIAL SOLUTION:

Statement 1 is not sufficient, as B could be the square of a very large prime number (71-squared only has 3 factors, but it's a giant number) while A could be a small number with several factors (16 has five factors, for example), providing the answer "no". Or B could be 1 and A could be 2 (or any other combination that gets you to "yes").

Statement 2 is also not sufficient, as here A could equal B, or A could be B times one more factor (4 and 4, or 8 and 4).

Taken together, the statements seem to be sufficient, but remember that your goal on Data Sufficiency should be to "play devil's advocate". It's easy to get the answer "yes" (A = 8 and B = 2 satisfies that). But to get "no", you should think about exponential increases. If B were 64, it doesn't quite have that many factors because so many are duplicates as is just 2 to the 6th power. That means that B would have 7 factors (1, 2, 4, 8, 16, 32, and 64). And in this case, A could be 48, which has ten factors (1, 2, 3, 4, 6, 8, 12, 16, 24, 48). In this case, A is NOT greater than B, giving us a "no" to add to our "yes", and making the correct answer E.
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ans E...
1) it does not tell us anything about their value... a prime no say 61 may have lesser factors but still be >30,which has more factors..
insufficient..
2) it again tells us that the prime factor of A are more or same as B.. insufficient
combined insufficient... say A=2^5*3*5*17 and B=2*3*17^4... here a has more factors yet it is smaller
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Here we go----

Best approach to solve these type of problems - pick values

St1: A has more factors than B does.

A = 12 = 2^2 * 3 ---> Number of factors 6
B = 49 = 7^2 ----> Number of factors 3

But B > A

Clearly not sufficient.

St2: Every prime factor of B is a factor of A.

B = 12 ---> Prime Factors-> 2,3
A = 6 ----> Prime factors-> 2,3

But B > A

Combining

A has more factors than B does
Every prime factor of B is a factor of A

Number of factors are given by multipying the prime factors of the given number after adding 1 to prime factors.

Every prime factor of B is a factor of A, and A has more factors.

b = 2 * 3 * (5)^10
a = 2^(8) * 3^(2) * 5

But b > a

option E is correct

E

approach:
Question is A>B

from 1: A has more factors than B
let A = 12, B =13--> we end up with A having more factors than B
if A =13, B =12 --> B has more factors, 2 diff answers NSF

from 2:
try numbers:
A =13, B =69, we have A<B, still the prime factors of A and B are same.
if A = 20, B =10, then A> B and the condition still meets --> NSF

1+2:
still not sufficient as the number of factors does not tell anything on the size of the numbers from 1 (solution) and we still end up with case 2, which lead to 2 answers
NSF

E

How I got to E:

[1] A has more factors than B does.
Perhaps A has more factors, but these factors are of a small value.
E.g A=2*2*2 = 8
and B= 3*5 = 15

[2] Every prime factor of B is a factor of A.
To me this meant that A has all of the factors of B, plus some more.

So, if B=2*2*3, then A could be:
A=2*2*3*1. This would make A=B
A=2*2*3*1*5. This would make A>B.

For the same reasons [1] and [2] together are NS.

Bunuel, the lesson of this question could be that more factors of a positive integer than another positive integer, doesn't mean that first integer is greater that the latter one.
Also that the inclusion of all prime factors of a positive integer in another positive integer, doesn't mean that first integer is greater that the latter one.
Please correct me if I misunderstood.

Yes, for example: 6 has four factors but is less than 7, which has 2 factors.



Yes, it'll depend on powers of these primes.
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Answer E

(1) A has more factors than B does.
- In Suff, As A can be any composite number and B can be a prime number greater than A and have number of factors (1) less than that of A OR B can be a prime number less than A and still number of factors(1) less than that of A.

(2) Every prime factor of B is a factor of A.
- In Suff, just knowing the every prime factors of A and B does not tell us which is greater

(1) + (2) =>
Let common prime factors be \((5,7)\)

Let A : \(5^4 * 7^2\) => no of factors \((4+1)(2+1) = 15\)
B : \(5^5 * 7^1\) => no of factors\((5+1)(1+1) = 12\)
in this case, A > B

Let A: \(7^4 * 5^2\) => no of factors \((4+1)(2+1) = 15\)
B: \(7^5 * 5^1\) => no of factors \((5+1)(1+1) = 12\)
in this case, A < B

Insuff => Answer (E)

I took statement two to mean that EVERY prime factor is B is in A.
Which means that if B is 4 (2*2), then A is AT LEAST (2*2) since it has EVERY prime factor of B.

This would make the answer C. I get that maybe that's not the lesson being taught in the question, but I'm sure other people read it that way as well.

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