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# If p and q are integers, can (q − 1) always be expressed as an integer

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Math Expert
Joined: 02 Sep 2009
Posts: 54370
If p and q are integers, can (q − 1) always be expressed as an integer  [#permalink]

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30 Jul 2017, 22:37
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65% (hard)

Question Stats:

34% (01:47) correct 66% (01:20) wrong based on 47 sessions

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If p and q are integers, can (q − 1) always be expressed as an integer multiple of p?

(1) p > q
(2) q > 1

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Joined: 21 Mar 2016
Posts: 520
Re: If p and q are integers, can (q − 1) always be expressed as an integer  [#permalink]

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31 Jul 2017, 10:56
Bunuel wrote:
If p and q are integers, can (q − 1) always be expressed as an integer multiple of p?

(1) p > q
(2) q > 1

stat2: clearly not suff no info on p

stat1: p > q,, so obvious p >q-1

but what if q is 1,,then q-1 is 0 and it can expressed as integer multiple

both combined,,suff

ans C
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Joined: 11 May 2015
Posts: 34
Location: United States
Concentration: Strategy, Operations
GPA: 3.44
Re: If p and q are integers, can (q − 1) always be expressed as an integer  [#permalink]

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31 Jul 2017, 14:56
mohshu wrote:
Bunuel wrote:
If p and q are integers, can (q − 1) always be expressed as an integer multiple of p?

(1) p > q
(2) q > 1

stat2: clearly not suff no info on p

stat1: p > q,, so obvious p >q-1

but what if q is 1,,then q-1 is 0 and it can expressed as integer multiple

both combined,,suff

ans C

Isnt the answer B in that case
q - 1 is not 0 and cannot always be expressed as a multiple of p - Suff.
Intern
Joined: 21 Jun 2017
Posts: 10
GMAT 1: 750 Q50 V41
Re: If p and q are integers, can (q − 1) always be expressed as an integer  [#permalink]

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31 Jul 2017, 15:24
mohshu wrote:
Bunuel wrote:
If p and q are integers, can (q − 1) always be expressed as an integer multiple of p?

(1) p > q
(2) q > 1

stat2: clearly not suff no info on p

stat1: p > q,, so obvious p >q-1

but what if q is 1,,then q-1 is 0 and it can expressed as integer multiple

both combined,,suff

ans C

Isnt the answer B in that case
q - 1 is not 0 and cannot always be expressed as a multiple of p - Suff.

q can be 5 and p can be 4 --> yes
q can be 2 and p can be 6 --> no
q can be 5 and p can be 3 --> no
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Joined: 16 Apr 2017
Posts: 1
Re: If p and q are integers, can (q − 1) always be expressed as an integer  [#permalink]

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31 Jul 2017, 23:54
Bunuel wrote:
If p and q are integers, can (q − 1) always be expressed as an integer multiple of p?

(1) p > q
(2) q > 1

Statement 1(p>q)
Case-1 - Let p=5 and q=-4, then (q-1=-5 is (-1) times p or 5)
Case-2 - Let p=7 and q =5, then (q-1=4 cannot be expressed as a multiple of p or 7)
Hence, Statement 1 is not sufficient.

Statement 2
Case-1 - Let p=7 and q=8, then (q-1=7 is (1) times p or 7)
Case-2 - Let p=7 and q =9, then (q-1=8 cannot be expressed as a multiple of p or 7)
Hence, Statement 2 is not sufficient.

Both Statements together
Here q is positive and >1 & p is also positive and greater than 2 (since both p & q are intergers)
Thus, p is always greater than q-1, since p & q are positive and p>q.
A small number can never be expressed as an integer multiple of a large number.
Hence, both statements together are sufficient.
Intern
Joined: 09 Dec 2014
Posts: 37
Re: If p and q are integers, can (q − 1) always be expressed as an integer  [#permalink]

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01 Aug 2017, 02:25
Bunuel wrote:
If p and q are integers, can (q − 1) always be expressed as an integer multiple of p?

(1) p > q
(2) q > 1

We have to find whether it is possible to express (q-1) as an integer multiple of p ==> (q-1) = p * k (k is some integer multiple)

Statement 1 says p>q.
Let q= 1 and p= 4
1-1 = 0 ==> 4 * 0, for k=0 --> possible
Let q=2 and p=4
2-1 = 1 But 1 cannot be expressed as a product of 4 * k, for some integer k. --> not possible
Insufficient

Statement 2 says q>1
Nothing has been given about the value of p
Let q=2 and p=1
2-1= 1 ==> can be expressed as 1*1, for k=1 --> possible
Let q=2 and p=3
2-1=1 ==> cannot be expressed as a product of 3 * k, for some integer k --> not possible
Insufficient

On combining both statements we have.
p>q and q>1 ==> 1<q<p
As p>q ==> p>q-1
So q-1 cannot be expressed as p*k, for some integer k
Thus, we get a definite NO after combining both the statements.
_________________
Thanks,
Ramya
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Joined: 07 Jun 2017
Posts: 100
Re: If p and q are integers, can (q − 1) always be expressed as an integer  [#permalink]

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01 Aug 2017, 03:05
Bunuel wrote:
If p and q are integers, can (q − 1) always be expressed as an integer multiple of p?

(1) p > q
(2) q > 1

Because the question is asking "Always"
I can be wrong
Intern
Joined: 09 Dec 2014
Posts: 37
Re: If p and q are integers, can (q − 1) always be expressed as an integer  [#permalink]

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01 Aug 2017, 08:40
pclawong wrote:
Bunuel wrote:
If p and q are integers, can (q − 1) always be expressed as an integer multiple of p?

(1) p > q
(2) q > 1

Because the question is asking "Always"
I can be wrong

In order to say a statement is sufficient to answer a given question in the Yes/No type of DS question, that statement has to give you a definite YES or a definite NO. Yes, the statement is sufficient if it gives you a definite NO. But a statement is not sufficient if it gives you YES sometimes (to some values) and NO to some values. In this question, as you have also mentioned "always", each statement is giving YES and NO if you consider them individually. That is not "always". So D cannot be answer to this question as per my knowledge. If you solved in some other way and got definite YES or NO to each statement when considering individually, please share. May be I have committed some mistake while solving. After all, we learn more as we discuss.
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Thanks,
Ramya
Re: If p and q are integers, can (q − 1) always be expressed as an integer   [#permalink] 01 Aug 2017, 08:40
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