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If a and b are different positive integers and a+b=a(a+b)  [#permalink]

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24 00:00

Difficulty:   75% (hard)

Question Stats: 45% (01:26) correct 55% (01:35) wrong based on 574 sessions

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If a and b are different positive integers and a + b = a(a + b), then which of the following must be true?

I. a = 1
II. b = 1
III. a < b

A. I only
B. II only
C. III only
D. I and II
E. I and III
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Re: DS - Number theory  [#permalink]

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6
1
hemanthp wrote:
If a and b are different positive integers and a + b = a(a + b), then which of the following must be true?
I. a = 1
II. b = 1
III. a < b

A. I only
B. II only
C. III only
D. I and II
E. I and III

This is slightly tricky. Watch out! Don't get angry if your answer didn't match the OA. I want to have a discussion around this.
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$$a + b = a(a + b)$$ --> $$a(a+b)-(a+b)=0$$ --> $$(a+b)(a-1)=0$$ --> as $$a$$ and $$b$$ are positive the $$a+b\neq{0}$$, so $$a-1=0$$ --> $$a=1$$. Also as $$a$$ and $$b$$ are different positive integers then $$b$$ must be more than $$a=1$$ --> $$a<b$$ ($$b$$ can not be equal to $$a$$ as they are different and $$b$$ can not be less than $$a$$ as $$b$$ is positive integerand thus can not be less than 1).

So we have that: $$a=1$$ and $$a<b$$.

I. a = 1 --> true;
II. b = 1 --> not true;
III. a < b --> true.

Answer: E (I and II only).

Hope it's clear.
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Re: DS - Number theory  [#permalink]

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2
Awesome! This was the exact point I wanted to bring up. And I was hoping for Bunuel to reply to the post. Thanks dude.

A and B are distinct positive integers. Why can't B be 0? Isn't 0 a positive integer? It sure as hell isn't negative. And indeed WIKI says "An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive."

So per GMAT ..is zero positive or neither positive or negative?

Thank you again ..Bunuel.
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Re: DS - Number theory  [#permalink]

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hemanthp wrote:
Awesome! This was the exact point I wanted to bring up. And I was hoping for Bunuel to reply to the post. Thanks dude.

A and B are distinct positive integers. Why can't B be 0? Isn't 0 a positive integer? It sure as hell isn't negative. And indeed WIKI says "An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive."

So per GMAT ..is zero positive or neither positive or negative?

Thank you again ..Bunuel.

Zero is neither positive not negative integer.
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Re: DS - Number theory  [#permalink]

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1
I pick numbers for this problem

a+b = a(a+b). to make the statement true a=1,b=2
1+2=1(1+2)
3 =3.

from this we know ,a=1 & a<b. answer should be E
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Re: DS - Number theory  [#permalink]

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TomB wrote:
I pick numbers for this problem

a+b = a(a+b). to make the statement true a=1,b=2
1+2=1(1+2)
3 =3.

from this we know ,a=1 & a<b. answer should be E

Number picking might not be the best way to solve MUST BE TRUE questions.

The question asks which of the following MUST be true, or which of the following is ALWAYS true no matter what set of numbers you choose. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

So the set you chose just proves that II is not always true and hence it's not a part of a correct choice. As for I and III: they might be true for this particular set of numbers but not true for another set, so you can not say that I and III are always true just based on one set of numbers (it just happens to be that I and III are always true).

As for "COULD BE TRUE" questions:
The questions asking which of the following COULD be true are different: if you can prove that a statement is true for one particular set of numbers, it will mean that this statement could be true and hence is a correct answer.

Hope it helps.
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Re: DS - Number theory  [#permalink]

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tricky question, i went with A only to realize that both a and b are positive integers, so b must be greater than a
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Re: DS - Number theory  [#permalink]

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Tricky one indeed. I marked E but was wrong in solutioning. I thought a=-b means a is smaller than b. Didn't realise that this equation is not possible since both a and b are positive
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Re: DS - Number theory  [#permalink]

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a+b=a(a+b)
1(a+b)-a(a+b)=0
(1-a)*(a+b)=0 from this either a-1=0 or a+b=0, but since a and b are positive their could not be equal to 0.
Thus, 1-a=0 or a=1.
1. holds true
2.not true because a and b are different positive integers and we know that a=1
3. true- As a, b are "different positive integers" when 1 is the smallest , thus a<b.

hence E.
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Re: DS - Number theory  [#permalink]

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1 is the smallest positive integer.
0 is a signless integer. These two are required.

from the equation we get a=1 as (a+b) cannot be zero.
so
I. a = 1 always true
II. b = 1 not always true
III. a < b always true

so the ans is E
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If a and b are different positive integers and a + b = a(a + b),  [#permalink]

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$$a+b = a (a+b)$$
$$\frac{a+b}{a+b}=a$$
$$a=1$$

I. a = 1 always TRUE
II. b = 1 b must not be the same as a. Thus, b is not equal to 1. FALSE
III. a < b b is a positive integer but not equal to 1 then it must be 2 onwards. always TRUE

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Re: If a and b are different positive integers and a+b=a(a+b)  [#permalink]

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Re: If a and b are different positive integers and a+b=a(a+b)  [#permalink]

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hemanthp wrote:
If a and b are different positive integers and a + b = a(a + b), then which of the following must be true?

I. a = 1
II. b = 1
III. a < b

A. I only
B. II only
C. III only
D. I and II
E. I and III

we can rewrite the equation as:
(a+b)/a = a+b
a/a=1
a+b/a = a+b
we can subtract a from both sides
b/a=b
this is true only when A is 1.
since we are told that the numbers are different, and both positive, we know for sure that b MUST be >1, or b>a.

E
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Re: If a and b are different positive integers and a+b=a(a+b)  [#permalink]

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hemanthp wrote:
If a and b are different positive integers and a + b = a(a + b), then which of the following must be true?

I. a = 1
II. b = 1
III. a < b

A. I only
B. II only
C. III only
D. I and II
E. I and III

a + b = a(a + b)
=> (a+b)(a-1) = 0
=> a=-b , a =1

a and b are both positive. So, a = -b is not possible.

a=1,
Also a and b are both different positive integers. So, b >1.

and hence a<b

So, I & II both are correct.

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Re: If a and b are different positive integers and a+b=a(a+b)  [#permalink]

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hemanthp wrote:
If a and b are different positive integers and a + b = a(a + b), then which of the following must be true?

I. a = 1
II. b = 1
III. a < b

A. I only
B. II only
C. III only
D. I and II
E. I and III

Since a and b are positive integers, a + b ≠ 0. So we can divide the given equation by a + b and obtain:

1 = a

Therefore, Roman numeral I is true. Furthermore, since a and b are different positive integers, b > a. So Roman numeral III is also true.

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