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

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Math Expert
Joined: 02 Sep 2009
Posts: 49968
If a and b are positive integers, is ab < 6?  [#permalink]

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26 Sep 2016, 03:27
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35% (medium)

Question Stats:

69% (01:55) correct 31% (01:34) wrong based on 126 sessions

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

(1) 1 < a + b < 7

(2) ab = a + b

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Re: If a and b are positive integers, is ab < 6?  [#permalink]

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26 Sep 2016, 05:31
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Bunuel wrote:
If a and b are positive integers, is ab < 6?

(1) 1 < a + b < 7

(2) ab = a + b

Stat 1: a+b = 1+1 or 2+2 or 3+3...then ab can be 1,4 or 9 respectively....different values...Insufficient.

Stat 2: ab = a+b..

if both 1*1 = 2...incorrect

2*2 = 4...correct

3*3 = 6...incorrect...thus only for a and b = 2 only then we get correct case...so ab < 6...Sufficient.

IMO option B.
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Re: If a and b are positive integers, is ab < 6?  [#permalink]

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07 Jun 2017, 20:23
Given a & b are +ve integers lets assume a=4 and b=2
1<4+2<7
1<6<7 ..
ab=8<6 FALSE
now if a,b are 1 and 2 then ab=2<6 TRUE
hence I is not sufficient

2) ab=a+b
only when a=b=2
thus ab<2*2,6 TRUE

Only II is sufficient
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Re: If a and b are positive integers, is ab < 6?  [#permalink]

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22 Sep 2018, 15:56
Bunuel wrote:
If a and b are positive integers, is ab < 6?

(1) 1 < a + b < 7

(2) ab = a + b

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

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

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

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

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

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

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

Regards,
Fabio.
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Re: If a and b are positive integers, is ab < 6? &nbs [#permalink] 22 Sep 2018, 15:56
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