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# If the greatest common divisor of p and s is 6, which of the following

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If the greatest common divisor of p and s is 6, which of the following [#permalink]

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22 Sep 2016, 05:17
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If the greatest common divisor of p and s is 6, which of the following must be true?

I) s+p is divisible by 18
II) s+2p is divisible by 18
III) s·p is divisible by 18

A. I only
B. II only
C. III only
D. I and II only
E. II and III only
[Reveal] Spoiler: OA

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If the greatest common divisor of p and s is 6, which of the following [#permalink]

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22 Sep 2016, 06:38
Bunuel wrote:
If the greatest common divisor of p and s is 6, which of the following must be true?

I) s+p is divisible by 18
II) s+2p is divisible by 18
III) s·p is divisible by 18

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

1) S+P / 18 we need to know since S and P have GCD of 6 , we need to take numbers that are numbers of 6.

Now 6(S+P) / 18 = S+P/3 = if S = odd/even or P = odd/even...based on number we get yes case or no case..

For ex :3+2/3 - no 3+3 / 3 = Yes... Not true.

2) s+2p/ 3 = same like above yes or no case...not true.

3) S-P/ 3 = we can take S - odd/even P -odd/even and need to take factors of 3 and check s-p is divisible by 3.

S P s-p
36(even) 15(odd) Yes
36 36 yes
15 15 yes
15 36 yes...

Hence only option 3 satisfy the case...

IMO option C.

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If the greatest common divisor of p and s is 6, which of the following [#permalink]

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22 Sep 2016, 08:08
Quote:
If the greatest common divisor of p and s is 6,

p and q are multiple of 6. {6,12,18,24,30......}

I) s+p is divisible by 18

Let s=6 and p=6 ==> s+p = 12 -- Not divisible by 18.

II) s+2p is divisible by 18

Let s=6 and p=12 ==> s+p = 30 -- Not divisible by 18.

III) s·p is divisible by 18

Let s=6 and p =6 ==> s.p = 36 -- Divisible by 18. Also for any value of s & p greater than 6 their multiple will be divisible by 18.

Ans. C
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If the greatest common divisor of p and s is 6, which of the following [#permalink]

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23 Sep 2016, 05:34
Top Contributor
Bunuel wrote:
If the greatest common divisor of p and s is 6, which of the following must be true?

I) s+p is divisible by 18
II) s+2p is divisible by 18
III) s·p is divisible by 18

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

p and s must be multiple of 6.

I) when p=6,s=12,then s+p=18 is divisible by 18.But when p=6,s=24,p+s=30 is NOT divisible by 18
II) when p=6,s=6,then s+2p=18 is divisible by 18.But when p=6,s=12,p+s=20 is NOT divisible by 18
III) since lowest value of p and s is 6 ,lowest s*p=36,is divisible by 18.So all the multiple of p*s must be divisible by 18

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Last edited by AbdurRakib on 24 Sep 2016, 11:38, edited 1 time in total.

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Re: If the greatest common divisor of p and s is 6, which of the following [#permalink]

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23 Sep 2016, 21:08
AbdurRakib

Quote:
I) when p=6,s=6,then s+p=12 is divisible by 18.But when p=6,s=24,p+s=30 is NOT divisible by 18

12 is not divisible by 18.

$$\frac{12}{18}=\frac{(2*2*3)}{(2*3*3)} =\frac{2}{3}$$
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Re: If the greatest common divisor of p and s is 6, which of the following [#permalink]

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24 Sep 2016, 11:40
Top Contributor
0ld wrote:
AbdurRakib

Quote:
I) when p=6,s=6,then s+p=12 is divisible by 18.But when p=6,s=24,p+s=30 is NOT divisible by 18

12 is not divisible by 18.

$$\frac{12}{18}=\frac{(2*2*3)}{(2*3*3)} =\frac{2}{3}$$

Thanks for mentioning me.

I already corrected the written error
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Re: If the greatest common divisor of p and s is 6, which of the following [#permalink]

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24 Sep 2016, 19:17
I) s+p is divisible by 18
II) s+2p is divisible by 18
III) s·p is divisible by 18

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

I first started w/ p=6 and s=12. From there I got that I and III were valid. Then I tried p=6 and s=18. From there I found that only statement 3 was valid. C.

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Re: If the greatest common divisor of p and s is 6, which of the following [#permalink]

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29 Apr 2017, 18:16
Fabulous Question testing the GCD-LCM knowledge.

I used test cases.

Option 1 =>
s=6
p=6

REJECTED

Option 2 =>

s=6
p=18

REJECTED.

Option 3 =>
As GCD=6
Hence both s and p must have a 6.
So sp will be divisible by 18.

Accepted.

SMASH THAT C.

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Re: If the greatest common divisor of p and s is 6, which of the following [#permalink]

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01 May 2017, 14:17
I chose two sets of numbers to test each case: {1} s=6, p=12 and {2} s=12, p=18.

Case 1: s+p is divis by 18. True in {1}, not true in {2}
Case 2: s+2p is divis by 18. True in {2}, not true in {1}.
Right here we can eliminate A, B, D, and E. I tested C just in case though.
Case 3: s*p is true in both {1} and {2}, because s and p will contribute the factors necessary to reach divisibility by 18.

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If the greatest common divisor of p and s is 6, which of the following [#permalink]

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05 Jul 2017, 21:23
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Bunuel wrote:
If the greatest common divisor of p and s is 6, which of the following must be true?

I) s+p is divisible by 18
II) s+2p is divisible by 18
III) s·p is divisible by 18

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

Because "greatest common divisor of p and s is 6", p= 6m and s=6n (m, n are positive integers)

s+p= 6(m+n).
If (m+n) is a multiple of 3, e.g. (m+n)=3, then (s+p) is divisible by 18
If (m+n) is not a multiple of 3, e.g. (m+n)=5, then (s+p) is not divisible by 18
--> Inconsistent answers --> (I) is not a must
Eliminate options (A), (D)

s+2p= 6(m+2n).
Similar to the above reasoning, (m+2n) could be either a multiple of 3 or not a multiple of 3
--> Inconsistent answers --> (II) is not a must
Eliminate options (B), (E)

Only (III) left, and there is no kind of "all the above choices are wrong" option, so option (C) is the winner.

For assessment on statement (III)
s.p=6m x 6n= 36mn. 36 is already divisible by 18, so 36mn is too.
--> (III) must be true.

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Re: If the greatest common divisor of p and s is 6, which of the following [#permalink]

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16 Aug 2017, 08:11
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Bunuel wrote:
If the greatest common divisor of p and s is 6, which of the following must be true?

I) s+p is divisible by 18
II) s+2p is divisible by 18
III) s·p is divisible by 18

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

Since GCD (P,S) is 6
Let S=6a & P=6b where a,b are +ve integers
Therefore
S+P = 6a+6b =6(a+b)
S+2P = 6a+12b=6(a+2b) &
S*P = 6a*6b =36ab

In the light of above deductions
I) (S+P)/18=6(a+b)/18= Integer if (a+b) is a multiple of 3 so NOT TRUE always
II)(S+2P)/18=6(a+2b)/18= Integer if (a+2b) is a multiple of 3 so NOT TRUE always
III)(S*P)/18=36(ab)/18= Integer ALWAYS TRUE irrespective of the value of a,b

Therefore option C

Thanks
Dinesh

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Re: If the greatest common divisor of p and s is 6, which of the following [#permalink]

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17 Aug 2017, 05:47
Bunuel, I need help here.

In such questions, Can I take a number which is multiple of 6.
Like in here, (6 & 60 )
And then perform the operations on them as given?

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Re: If the greatest common divisor of p and s is 6, which of the following [#permalink]

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17 Aug 2017, 05:56
amitpandey25 wrote:
If the greatest common divisor of p and s is 6, which of the following must be true?

I) s+p is divisible by 18
II) s+2p is divisible by 18
III) s·p is divisible by 18

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

Bunuel, I need help here.

In such questions, Can I take a number which is multiple of 6.
Like in here, (6 & 60 )
And then perform the operations on them as given?

We are given that the greatest common divisor of p and s is 6, so yes, p = 6 and s = 60 is one of the possible cases. Generally the greatest common divisor of p and s is 6, means that p = 6m and s = 6n, where m and n are co-prime integers (integers, which do not share any common factors but 1).

Notice though that the question asks which of the following MUST be true, not COULD be true. So, one possible set might not be enough. For example, p = 6 and s = 6 is possible, which will make II true, but II is true only for some of the values (so it could be true) but it won't be true for all cases (so it's not always true) while III is always true, it's true for any (all) possible cases (so III must be true).

Hope it's clear.
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Re: If the greatest common divisor of p and s is 6, which of the following [#permalink]

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17 Aug 2017, 08:14
GCD(p,s) = 6
p,s are 6, multiple of 6
so take 6,6 for eg:
1) p+s=12 not divisible by 18.
2) p+2s=18 now take p=6,s=12 , p+2s = 30 not divisible by 18
3) p*s , take 6, any multiple of 6 their multiplication will be divisible by 18.

therefore C is correct
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Re: If the greatest common divisor of p and s is 6, which of the following   [#permalink] 17 Aug 2017, 08:14
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