If a and b are positive integers, is ab < 6?

\(a,b \geqslant 1\,\,{\text{ints}}\,\,\,\left( * \right)\)

\({\text{ab}}\,\,\mathop < \limits^? \,\,6\)

\(\left( 1 \right)\,\,1 < a + b < 7\,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {1,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr \\
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {2,3} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.\)


\(\left( 2 \right)\,\,ab = a + b\,\,\,\, \Rightarrow \,\,\,\,a\left( {b - 1} \right) = b\,\,\,\,\left( {**} \right)\)

\(a = 1\,\,\,{\text{OR}}\,\,\,b = 1\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,{\text{impossible}}\)

\(a,b\,\, \ge {\rm{2}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\left\{ \matrix{\\
a\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{divisor}}\,\,{\rm{of}}\,\,b\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)\,\,} \,\,\,\,\,a \le b\, \hfill \cr \\
b - 1\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{divisor}}\,\,{\rm{of}}\,\,b\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)\,\,} \,\,\,\,\,b = 2\, \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

If a and b are positive integers, is ab < 6?

(1) 1 < a + b < 7

(2) ab = a + b


A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;


B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;


C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the
question asked;


D. EACH statement ALONE is sufficient to answer the question asked;


E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Source : gmat.london.edu

Rephrasing the question

Is a=1 and b=1,2,3,4,5, or
Is a=2 and b=1,2

(1) 1 < a + b < 7
There are many possibilities,
Insufficient

(2) ab = a + b
this means both are equal
\(a^2\) = 2a
Since we know that both are +ve integers we can directly eliminate a from both sides
And, a = 2
so, ab = 4
Sufficient.

Option B

a and b are positive integers -- we know this from the question stem.

1) sum of a and b is between 1 and 7
Let's take two integers between 1 and 7--- 3 and 6
Now 3= a+b can be 1+2 --> In this case, ab=2 which is less than 6 -- NO
Now 6= a+b can be 3+3 --> In this case, ab=9 which is greater than 6 -- YES
Hence insuffiicient

2) ab= a+b

When sum of two positive integers is equal to their products, it means that the two integeral values are equal
We can rewrite it as:

\(a*a= a+a\)
\(a^2=2a\)

rearranging

\(a (a-2)=0\)

Since we know the two integers are positive, we can ignore a=0

Hence we get our answer as a=2

Sufficient
Hence answer is (B)

If a and b are positive integers, is ab < 6?

(1) 1 < a + b < 7

(2) ab = a + b

Solution:
The stem implies one of the following possibilities:
1. a and b are both positive (so ab could be less than 6, or greater than 6)
2. a and b are both negative (so ab could be less than 6, or greater than 6)
3. a and b share opposite signs (means ab is definitively less than 6)

(1) 1 < a + b < 7
This statement implies that a and b could be 1 above or 3.
Insufficient.

(2) ab = a + b
This implies that a and b must both be equal to 2.
Sufficient.

While your answer is correct, I think you missed the part in the stem where its given that a and b are positive integers and your 2nd and 3rd point of stem evaluation are incorrect. Not the most fatal mistake for this particular question, but stuff like that can be deadly at times!

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