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M05-22

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Official Solution:

If x is a positive integer, is x divisible by 15?

(1) \(x\) is a multiple of 10. If \(x=10\) then the answer is NO but if \(x=30\) then the answer is YES. Not sufficient.

(2) \(x^2\) is a multiple of 12. The least perfect square which is a multiple of 12 is 36. Hence, the least value of \(x\) is 6 and in this case the answer is NO, but if for example \(x=12*15\) then the answer is YES. Not sufficient.

Notice that from this statement we can deduce that \(x\) must be a multiple of 3 (else how can this prime appear in \(x^2\)?).

(1)+(2) \(x\) is a multiple of both 10 and 3, hence it's a multiple of 30, so \(x\) must be divisible by 15. Sufficient.


Answer: C
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Re: M05-22 [#permalink]

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New post 27 Sep 2014, 12:20
Bunuel wrote:
Official Solution:


(1) \(x\) is a multiple of 10. If \(x=10\) then the answer is NO but if \(x=30\) then the answer is YES. Not sufficient.

(2) \(x^2\) is a multiple of 12. The least perfect square which is a multiple of 12 is 36. Hence, the least value of \(x\) is 6 and in this case the answer is NO, but if for example \(x=12*15\) then the answer is YES. Not sufficient.

Notice that from this statement we can deduce that \(x\) must be a multiple of 3 (else how can this prime appear in \(x^2\)?).

(1)+(2) \(x\) is a multiple of both 10 and 3, hence it's a multiple of 30, so \(x\) must be divisible by 15. Sufficient.


Answer: C



I couldn't follow the highlighted portion actually.

I seem to find DS Questions with given statements such as below one little difficult/time consuming:
1) x multiple of 55
2) x^3 multiple of 110

Please suggest if there are any familiar types your aware of.

Would be really grateful.

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New post 27 Sep 2014, 12:45
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earnit wrote:
Bunuel wrote:
Official Solution:


(1) \(x\) is a multiple of 10. If \(x=10\) then the answer is NO but if \(x=30\) then the answer is YES. Not sufficient.

(2) \(x^2\) is a multiple of 12. The least perfect square which is a multiple of 12 is 36. Hence, the least value of \(x\) is 6 and in this case the answer is NO, but if for example \(x=12*15\) then the answer is YES. Not sufficient.

Notice that from this statement we can deduce that \(x\) must be a multiple of 3 (else how can this prime appear in \(x^2\)?).

(1)+(2) \(x\) is a multiple of both 10 and 3, hence it's a multiple of 30, so \(x\) must be divisible by 15. Sufficient.


Answer: C



I couldn't follow the highlighted portion actually.

I seem to find DS Questions with given statements such as below one little difficult/time consuming:
1) x multiple of 55
2) x^3 multiple of 110

Please suggest if there are any familiar types your aware of.

Would be really grateful.


Important thing to know when solving this problem is that exponentiation does not "produce" primes. For example, for integer x if x^(positive integer) is divisible by 3, then x must be divisible by 3. How else would 3 appear in x^(positive integer)?

We are given that \(x^2\), where x is an integer, is a multiple of 12 = 2^2*3.This means that x must be a multiple of both 2 and 3. If they weren't how can x^2 have these primes?

Similar questions to practice:
if-x-is-an-integer-is-x-3-divisible-by-165973.html
how-many-different-prime-numbers-are-factors-of-the-positive-126744.html
if-n-2-n-yields-an-integer-greater-than-0-is-n-divisible-by-126648.html
if-k-is-a-positive-integer-is-k-the-square-of-an-integer-55987.html
if-k-is-a-positive-integer-how-many-different-prime-numbers-95585.html
if-n-is-the-integer-whether-30-is-a-factor-of-n-126572.html
if-x-is-an-integer-is-x-2-1-x-5-an-even-number-104275.html

Hope it helps.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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What are GMAT Club Tests?
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Re: M05-22 [#permalink]

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New post 27 Sep 2014, 12:56
Bunuel wrote:
earnit wrote:
Bunuel wrote:
Official Solution:


(1) \(x\) is a multiple of 10. If \(x=10\) then the answer is NO but if \(x=30\) then the answer is YES. Not sufficient.

(2) \(x^2\) is a multiple of 12. The least perfect square which is a multiple of 12 is 36. Hence, the least value of \(x\) is 6 and in this case the answer is NO, but if for example \(x=12*15\) then the answer is YES. Not sufficient.

Notice that from this statement we can deduce that \(x\) must be a multiple of 3 (else how can this prime appear in \(x^2\)?).

(1)+(2) \(x\) is a multiple of both 10 and 3, hence it's a multiple of 30, so \(x\) must be divisible by 15. Sufficient.


Answer: C



I couldn't follow the highlighted portion actually.

I seem to find DS Questions with given statements such as below one little difficult/time consuming:
1) x multiple of 55
2) x^3 multiple of 110

Please suggest if there are any familiar types your aware of.

Would be really grateful.


Important thing to know when solving this problem is that exponentiation does not "produce" primes. For example, for integer x if x^(positive integer) is divisible by 3, then x must be divisible by 3. How else would 3 appear in x^(positive integer)?

We are given that \(x^2\), where x is an integer, is a multiple of 12 = 2^2*3.This means that x must be a multiple of both 2 and 3. If they weren't how can x^2 have these primes?

Similar questions to practice:
if-x-is-an-integer-is-x-3-divisible-by-165973.html
how-many-different-prime-numbers-are-factors-of-the-positive-126744.html
if-n-2-n-yields-an-integer-greater-than-0-is-n-divisible-by-126648.html
if-k-is-a-positive-integer-is-k-the-square-of-an-integer-55987.html
if-k-is-a-positive-integer-how-many-different-prime-numbers-95585.html
if-n-is-the-integer-whether-30-is-a-factor-of-n-126572.html
if-x-is-an-integer-is-x-2-1-x-5-an-even-number-104275.html

Hope it helps.


THANKS a ton.
Really appreciate it.
:)

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New post 17 Sep 2015, 07:24
Need great help , I don't understand why (2) can give both yes and no answer , many thanks
My understanding:
(2)X2 can be 36 or 144 which is x=6 or 12
Both are not divisible by 15
Many thsnks

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Re: M05-22 [#permalink]

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New post 28 Jun 2016, 19:17
Plz let me know if my approach is correct:

In order to know if x is divisible by 15, x has to contain at least the prime factors of \(15\) -> \((3*5)\).

From (1): We only know that x is similar to \(x*2*5\) ->\(x\) could be three. Not sufficient.

From (2): We only know that \(x^2\) is similar to \(x*x*2*2*3\) -> One of the \(x\) could be \(5\). Not sufficient.

From (1) + (2): We know that \(x\) contains at least one 3 and one 5 what makes it divisible by \(15\) -> \((3*5)\). Sufficient

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New post 28 Jun 2016, 23:29
paddy41 wrote:
Plz let me know if my approach is correct:

In order to know if x is divisible by 15, x has to contain at least the prime factors of \(15\) -> \((3*5)\).

From (1): We only know that x is similar to \(x*2*5\) ->\(x\) could be three. Not sufficient.

From (2): We only know that \(x^2\) is similar to \(x*x*2*2*3\) -> One of the \(x\) could be \(5\). Not sufficient.

From (1) + (2): We know that \(x\) contains at least one 3 and one 5 what makes it divisible by \(15\) -> \((3*5)\). Sufficient

______________
Yes, that's correct.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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M05-22 [#permalink]

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New post 06 May 2017, 20:46
Bunuel wrote:
Important thing to know when solving this problem is that exponentiation does not "produce" primes. For example, for integer x if x^(positive integer) is divisible by 3, then x must be divisible by 3. How else would 3 appear in x^(positive integer)?

We are given that \(x^2\), where x is an integer, is a multiple of 12 = 2^2*3.This means that x must be a multiple of both 2 and 3. If they weren't how can x^2 have these primes?


I don't think I understand this. Are you saying that if \(x^2\) is divisible by 12 then x is divisible by 12?

So does that mean that if \(x^z\) is divisible by y then x is divisible by y (as long as z is positive)?

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Re: M05-22 [#permalink]

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New post 07 May 2017, 00:16
joondez wrote:
Bunuel wrote:
Important thing to know when solving this problem is that exponentiation does not "produce" primes. For example, for integer x if x^(positive integer) is divisible by 3, then x must be divisible by 3. How else would 3 appear in x^(positive integer)?

We are given that \(x^2\), where x is an integer, is a multiple of 12 = 2^2*3.This means that x must be a multiple of both 2 and 3. If they weren't how can x^2 have these primes?



So does that mean that if \(x^z\) is divisible by y then x is divisible by y (as long as z is positive)?


...as long as y is prime your statement should be correct.

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New post 07 May 2017, 01:03
paddy41 wrote:
joondez wrote:
Bunuel wrote:
Important thing to know when solving this problem is that exponentiation does not "produce" primes. For example, for integer x if x^(positive integer) is divisible by 3, then x must be divisible by 3. How else would 3 appear in x^(positive integer)?

We are given that \(x^2\), where x is an integer, is a multiple of 12 = 2^2*3.This means that x must be a multiple of both 2 and 3. If they weren't how can x^2 have these primes?



So does that mean that if \(x^z\) is divisible by y then x is divisible by y (as long as z is positive)?


...as long as y is prime your statement should be correct.


You should read more carefully. As correctly noted by paddy41, the property above is talking about primes.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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New post 13 Oct 2017, 03:57
how sqrt(180) gives a integer??? we cant take 180 has a sol..

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Re: M05-22   [#permalink] 13 Oct 2017, 05:37
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