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# If 90/k is an integer, is k an integer?

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Re: If 90/k is an integer, is k an integer? [#permalink]
enigma123 wrote:
If 90/k is an integer, is k an integer?

1. k>1
2. k is a multiple of a prime number.

My approach:

Considering Statement 1

k>1 so k can be an integer or could be in decimals i.e. 1.5. So insufficient

Considering Statement 2

K can be 2,3, 5, 9, --Therefore K has to be integer and sufficient.

Again is my concept and approach correct in answering this question?

Statement 2 ->
K is a multiple of prime number = > 1.5 = 3 * .5 ( 3 is a prime no. K = 1.5 still holds)

Combining both statements still 1.5 holds and 3 holds true so E is the answer. Am I correct, what am I doing wrong ?

Regards,
wikdwik
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Re: If 90/k is an integer, is k an integer? [#permalink]
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wikdwik wrote:
enigma123 wrote:
If 90/k is an integer, is k an integer?

1. k>1
2. k is a multiple of a prime number.

My approach:

Considering Statement 1

k>1 so k can be an integer or could be in decimals i.e. 1.5. So insufficient

Considering Statement 2

K can be 2,3, 5, 9, --Therefore K has to be integer and sufficient.

Again is my concept and approach correct in answering this question?

Statement 2 ->
K is a multiple of prime number = > 1.5 = 3 * .5 ( 3 is a prime no. K = 1.5 still holds)

Combining both statements still 1.5 holds and 3 holds true so E is the answer. Am I correct, what am I doing wrong ?

Regards,
wikdwik

On the GMAT when we are told that $$a$$ is divisible by $$b$$ (or which is the same: "$$a$$ is multiple of $$b$$", or "$$b$$ is a factor of $$a$$"), we can say that:
1. $$a$$ is an integer;
2. $$b$$ is an integer;
3. $$\frac{a}{b}=integer$$.

So, k is a multiple of prime number means that k = prime*integer, which basically implies that k can be any integer but 1.
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Re: If 90/k is an integer, is k an integer? [#permalink]
Bunuel wrote:
wikdwik wrote:
enigma123 wrote:
If 90/k is an integer, is k an integer?

1. k>1
2. k is a multiple of a prime number.

My approach:

Considering Statement 1

k>1 so k can be an integer or could be in decimals i.e. 1.5. So insufficient

Considering Statement 2

K can be 2,3, 5, 9, --Therefore K has to be integer and sufficient.

Again is my concept and approach correct in answering this question?

Statement 2 ->
K is a multiple of prime number = > 1.5 = 3 * .5 ( 3 is a prime no. K = 1.5 still holds)

Combining both statements still 1.5 holds and 3 holds true so E is the answer. Am I correct, what am I doing wrong ?

Regards,
wikdwik

On the GMAT when we are told that $$a$$ is divisible by $$b$$ (or which is the same: "$$a$$ is multiple of $$b$$", or "$$b$$ is a factor of $$a$$"), we can say that:
1. $$a$$ is an integer;
2. $$b$$ is an integer;
3. $$\frac{a}{b}=integer$$.

So, k is a multiple of prime number means that k = prime*integer, which basically implies that k can be any integer but 1.

When you say on the GMAT, does it means that it will not follow the general logic? And for Prime Numbers, can this be accepted as a TRUTH for Prime Numbers, that if any Number (N) is divided by any other INTEGER (K), which is also a Multiple of a Prime Number (All of them are), results in an INTEGER, this will imply that K will always be Integer?

too much to remeber
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Re: If 90/k is an integer, is k an integer? [#permalink]
sandeepmanocha wrote:
When you say on the GMAT, does it means that it will not follow the general logic? And for Prime Numbers, can this be accepted as a TRUTH for Prime Numbers, that if any Number (N) is divided by any other INTEGER (K), which is also a Multiple of a Prime Number (All of them are), results in an INTEGER, this will imply that K will always be Integer?

too much to remeber

I tried to understand what you've written there but failed...
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Re: If 90/k is an integer, is k an integer? [#permalink]
clearly statement 1 not enough -

For statement 2 alone -
K is multiple of prime number means it could be -
2 4 6 8 10 ...
3 6 9 12 15 ...
5 10 15 20 25 ...
7 14 21 ...

now if we factorize 90 = 3*3*5*2
so in short 3*3*5*2 / k should be integer.

if we start inserting values - for K as mulitple of 2 , 3*3*5*2/4 is not integer
for K as multiple of 3 , 3*3*5*2/12 is not integer
for k as multiple of 5 , 3*3*5*2/20 is not integer
for K as multiple of 7 and higher clearly its not integer ..

So both statement toghether not enough answer E
Where i m going wrong ???

Bunuel
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Re: If 90/k is an integer, is k an integer? [#permalink]
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clearly statement 1 not enough -

For statement 2 alone -
K is multiple of prime number means it could be -
2 4 6 8 10 ...
3 6 9 12 15 ...
5 10 15 20 25 ...
7 14 21 ...

now if we factorize 90 = 3*3*5*2
so in short 3*3*5*2 / k should be integer.

if we start inserting values - for K as mulitple of 2 , 3*3*5*2/4 is not integer
for K as multiple of 3 , 3*3*5*2/12 is not integer
for k as multiple of 5 , 3*3*5*2/20 is not integer
for K as multiple of 7 and higher clearly its not integer ..

So both statement toghether not enough answer E
Where i m going wrong ???

Bunuel

Question asks: is k an integer?

(2 ) says that k is a multiple of a prime number, which means that k IS an integer.
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If 90/k is an integer, is k an integer? [#permalink]
Sorry for the confusion - I will try to elaborate now with examples -

Bunuel wrote:
sandeepmanocha wrote:
When you say on the GMAT, does it means that it will not follow the general logic? And for Prime Numbers, can this be accepted as a TRUTH for Prime Numbers, that if any Number (N) is divided by any other INTEGER (K), which is also a Multiple of a Prime Number (All of them are), results in an INTEGER, this will imply that K will always be Integer?

too much to remeber

I tried to understand what you've written there but failed...

You said
Quote:
On the GMAT when we are told that a is divisible by b (or which is the same: "a is multiple of b", or "b is a factor of a"), we can say that:
1. a is an integer;
2. b is an integer;
3. ab=integer.

So, k is a multiple of prime number means that k = prime*integer, which basically implies that k can be any integer but 1.

But there is Another Possibility, like one given below

wikdwik wrote:

Statement 2 ->
K is a multiple of prime number = > 1.5 = 3 * .5 ( 3 is a prime no. K = 1.5 still holds)

Combining both statements still 1.5 holds and 3 holds true so E is the answer. Am I correct, what am I doing wrong ?

Now if I go by wikdwik my answer is E but if I go by your answer which is based on the concept "On the GMAT when we are told that a is divisible by...." then answer choice will be D

My question was - For the GMAT do we need to ALWAYS consider if the INTEGER (N) divided by any other NUMBER (K) = INTEGER, then K is also an INTEGER

Although following is also a possibility -
N=90
K = 4.5 = 9 * .5 (Not INTEGER)
N/K = 20 (Integer)
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Re: If 90/k is an integer, is k an integer? [#permalink]
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sandeepmanocha wrote:
Sorry for the confusion - I will try to elaborate now with examples -

Bunuel wrote:
sandeepmanocha wrote:
When you say on the GMAT, does it means that it will not follow the general logic? And for Prime Numbers, can this be accepted as a TRUTH for Prime Numbers, that if any Number (N) is divided by any other INTEGER (K), which is also a Multiple of a Prime Number (All of them are), results in an INTEGER, this will imply that K will always be Integer?

too much to remeber

I tried to understand what you've written there but failed...

You said
Quote:
On the GMAT when we are told that a is divisible by b (or which is the same: "a is multiple of b", or "b is a factor of a"), we can say that:
1. a is an integer;
2. b is an integer;
3. ab=integer.

So, k is a multiple of prime number means that k = prime*integer, which basically implies that k can be any integer but 1.

But there is Another Possibility, like one given below

wikdwik wrote:

Statement 2 ->
K is a multiple of prime number = > 1.5 = 3 * .5 ( 3 is a prime no. K = 1.5 still holds)

Combining both statements still 1.5 holds and 3 holds true so E is the answer. Am I correct, what am I doing wrong ?

Now if I go by wikdwik my answer is E but if I go by your answer which is based on the concept "On the GMAT when we are told that a is divisible by...." then answer choice will be D

My question was - For the GMAT do we need to ALWAYS consider if the INTEGER (N) divided by any other NUMBER (K) = INTEGER, then K is also an INTEGER

Although following is also a possibility -
N=90
K = 4.5 = 9 * .5 (Not INTEGER)
N/K = 20 (Integer)

Again, if k is NOT an integer it makes NO SENSE to say that k is a multiple of some integer.

If 90/k is an integer, is k an integer?

(1) k > 1. Not sufficient: k can be 2, so an integer or k can be 3/2, so not an integer.

(2) k is a multiple of a prime number. For k to be a multiple of an integer, it MUST be an integer, otherwise this statement won't have any sense. Sufficient.

Hope it's clear.
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If 90/k is an integer, is k an integer? [#permalink]
Hello, I worked on this one like this:

90 / k is an integer.
So, I tried t find the factors of 90, one of which should be k.

90....../.......Integer
90..................1
90..................2
45..................3
30..................4
x....................5
18..................6
15..................7
x....................8
x....................9
10.................10
9

[1] is not sufficient as all of the factors apart of 1 are greater than 1.

[2] is sufficient because if k is a multiple of a prime, taking primes and looking for their multiples we find that:
Taking 2, 2*15 = 30, k=30, 2*5=10, k = 10 etc. So, 90/k is an integer and k is an integer. Same in the cases below.
Taking 3, 3*10 = 30, k=30 etc
Taking 5, 5*2 = 10, k=10 etc
Taking 7, it gives none of the factors we found above, so it cannot be 7 (which we also see in the factor table above). Also, we know that 90/k is an integer, so this also rules out 7.
Taking 9, 9*10 = 90, k=90.

ANS is B

It looks long, but it actually took me 1.52 minute.
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Re: If 90/k is an integer, is k an integer? [#permalink]
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enigma123 wrote:
If 90/k is an integer, is k an integer?

(1) k > 1
(2) k is a multiple of a prime number.

Given: 90/k is an integer

Target question: Is k an integer?

Statement 1: k > 1
Let's TEST some values.
There are several values of k that satisfy statement 1. Here are two:
Case a: k = 10. Notice that 90/k = 90/10 = 9, which is an integer. In this case, the answer to the target question is YES, k IS an integer
Case b: k = 90/77. Notice that 90/k = 90/(90/77) = 90(77/90) = 77, which is an integer. In this case, the answer to the target question is NO, k is NOT an integer
ince we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: k is a multiple of a prime number.
All prime numbers are INTEGERS, and all multiples of integers are integers.
So, the answer to the target question is YES, k IS an integer
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
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Re: If 90/k is an integer, is k an integer? [#permalink]
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Re: If 90/k is an integer, is k an integer? [#permalink]
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