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If p and q are distinct integers, is 4 a factor of p – q?

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If p and q are distinct integers, is 4 a factor of p – q?  [#permalink]

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New post 06 Nov 2016, 09:11
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A
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C
D
E

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Re: If p and q are distinct integers, is 4 a factor of p – q?  [#permalink]

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New post 06 Nov 2016, 09:19
Here is my approach =>
Here we need to see whether p-q is divisible by 4 or not.
Statement 1=>
4 is a factor of p.
p/4=integer
In other words=>
p is divisible by 4
Lets use test cases=>
p=4
q=2 =>p-q=2 Clearly not divisible by 4
p=16
q=12
p-q=4 Clearly divisibly be 4

Hence insufficient
Statement 2
Q/4=integer.
Lets use the test cases again
q=4
p=8
p-q=4=> Clearly divisible by 4
q=4
p=5
p-q=1=>clearly not divisible by 4
Hence Insufficient.
Combining them
we can say that p=4x
and q=4y
for some integer x and y
hence p-q=4(x-y) => It is divisible by 4

Alternatively we could use the property => When we add (or subtract) two multiples of n => We get a multiple of n
Hence C
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Re: If p and q are distinct integers, is 4 a factor of p – q?  [#permalink]

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New post 06 Nov 2016, 09:22
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Bunuel wrote:
If p and q are distinct integers, is 4 a factor of p – q?

(1) 4 is a factor of p.
(2) 4 is a factor of q.


Target question: Is 4 a factor of p – q?

Statement 1: 4 is a factor of p
Since we don't have any information about q, there's now way to tell whether 4 is factor of p – q
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 4 is a factor of q
Since we don't have any information about p, there's now way to tell whether 4 is factor of p – q
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1: 4 is a factor of p. This means we can write p = 4k, for some integer k
Statement 2: 4 is a factor of q. This means we can write q = 4j, for some integer j
So, p - q = 4k - 4j = 4(k - j)
From this, we can see that p - q is equal to the product of 4 and some integer (k-j)
So, 4 must be a factor of p – q
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer:

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If p and q are distinct integers, is 4 a factor of p – q?  [#permalink]

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New post 06 Nov 2016, 10:03
Bunuel wrote:
If p and q are distinct integers, is 4 a factor of p – q?

(1) 4 is a factor of p.
(2) 4 is a factor of q.

There can be multiple ways to approach this question , I will try to approach this question by plugging in some number...

FROM STATEMENT - I ( INSUFFICIENT)

Let p = 8 & q = 2

p - q = 6 ; Not divisible by 4

Again -

Let P = 16 & q = 8

p - q = 8 ; Divisible by 4

Thus no unique solution can be obtained....


FROM STATEMENT - II ( INSUFFICIENT)


Let p = 16 & q = 8

p - q = 8 ; Divisible by 4

Again -

Let P = 14 & q = 8

p - q = 6 ; Not divisible by 4

Thus no unique solution can be obtained....


FROM STATEMENT - I & II ( SUFFICIENT)

When both p & q are multiples of 4 , 4 can be common factor and thus p - q will be divisible by 4

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

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Re: If p and q are distinct integers, is 4 a factor of p – q?  [#permalink]

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New post 06 Nov 2016, 23:23
Bunuel wrote:
If p and q are distinct integers, is 4 a factor of p – q?

(1) 4 is a factor of p.
(2) 4 is a factor of q.


IMO C
p and q are distinct integers

From statement 1
4 is a factor of p i.e., p is divisible by 4
but no information is given regarding q.
p can be 12 and q can be 1 in which case the answer is NO
p can be 12 and q can be 4 in which case the answer is YES
Hence insufficient

From statement 2
4 is a factor of q i.e., q is divisible by 4
no information is given regarding p.
hence insufficient (following similar discussion as in statement 1)

Combining
p and q are divisible by 4 giving p-q is divisible by 4
p=16 q=4 p-q=12 or p=4 q=12 p-q=-8 (both cases divisible by 4)
Hence C
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If p and q are distinct integers, is 4 a factor of p – q?  [#permalink]

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New post 27 Nov 2016, 01:11
from statement 1 :
p=4*a , but we don't have any information about q
not sufficient

from statement 2 :
q=4b, but we don't have any information about p
not sufficient

combining 1 and 2 :

p-q = 4a- 4b = 4(a-b)

irrespective of the values of a and b, 4 will always be a factor of p-q

answer is C
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If p and q are distinct integers, is 4 a factor of p – q?  [#permalink]

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New post 31 Aug 2018, 08:04
Bunuel wrote:
If p and q are distinct integers, is 4 a factor of p – q?

(1) 4 is a factor of p.
(2) 4 is a factor of q.

\(p \ne q\,\,\,{\text{ints}}\)

\(\frac{{p - q}}{4}\,\,\,\mathop = \limits^? \,\,\,\operatorname{int}\)

We will prove that each statement ALONE is insufficient to answer the question asked (in a unique way), through what we call an ALGEBRAIC BIFURCATION:

\(\left( 1 \right)\,\,\,\,\frac{p}{4} = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,\,Take\,\,\left( {p,q} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\,\,Take\,\,\left( {p,q} \right) = \left( {0,4} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\)

\(\left( 2 \right)\,\,\,\,\frac{q}{4} = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,\,Take\,\,\left( {p,q} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\,\,Take\,\,\left( {p,q} \right) = \left( {4,0} \right)\,\,\,\, \Rightarrow \,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\)

\(\left( {1 + 2} \right)\,\,\,\,\frac{{p - q}}{4} = \frac{p}{4} - \frac{q}{4} = \operatorname{int} - \operatorname{int} = \operatorname{int} \,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\)

The above follows the notations and rationale taught in the GMATH method.
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If p and q are distinct integers, is 4 a factor of p – q?   [#permalink] 31 Aug 2018, 08:04
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