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How many positive integers less than 250 are multiple of 4 but NOT mul

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Math Expert
Joined: 02 Sep 2009
Posts: 46162
How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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17 Apr 2015, 05:06
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67% (01:17) correct 33% (01:37) wrong based on 432 sessions

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How many positive integers less than 250 are multiple of 4 but NOT multiples of 6?

(A) 20
(B) 31
(C) 42
(D) 53
(E) 64

Kudos for a correct solution.

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Re: How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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17 Apr 2015, 05:18
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Bunuel wrote:
How many positive integers less than 250 are multiple of 4 but NOT multiples of 6?

(A) 20
(B) 31
(C) 42
(D) 53
(E) 64

Kudos for a correct solution.

248/4=62

Multiples of 4 which are a multiple of 6 will be of the form 2*2*3=12n where n>0
240/12=20

62-20=42

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Joined: 11 Sep 2013
Posts: 21
GMAT 1: 620 Q49 V27
Re: How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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18 Apr 2015, 13:49
1
Multiples of 4 less than 250 = {4,8,12,......248} = 62 numbers

Multiples of 4 which are multiples of 3 too ={12,24,36.....240} = 20 numbers

So required number= 62-20 = 42

Choice C
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Posts: 64
Re: How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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19 Apr 2015, 04:37
4
Positive integers multiple of 4 < 250 $$\,=\,\frac{248-4}{4}\,+1\,=\,62$$
Positive integers multiple of 12 (LCM of 4 & 6) < 250 $$\,=\,\frac{240-12}{12}\,+1\,=\,20$$

Positive integers < 250 multiple of 4 but NOT multiples of 6 $$\,=\,62\,-\,20\,=\,42$$

Or

Multiples of 4 <250 $$\,=\,\frac{250}{4}\,=\,62.5$$
In multiples of 4 <250 i.e. 4, 8, 12, 16, 20, 24.... every third term is multiple of 6

Positive integers < 250 multiple of 4 but NOT multiples of 6 $$\,=\,62.5\,-\,\frac{1}{3}\,*62.5$$
$$\,=\,62.5\,-20.8$$

$$\,=\,41.7\,\approx{42}$$

Math Expert
Joined: 02 Sep 2009
Posts: 46162
Re: How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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20 Apr 2015, 04:55
Bunuel wrote:
How many positive integers less than 250 are multiple of 4 but NOT multiples of 6?

(A) 20
(B) 31
(C) 42
(D) 53
(E) 64

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION

This is the type of question that most people can get with unlimited time. You can simply go through every possible number from 1 to 249 and see if each number meets the criteria. Apart from going cross-eyed halfway through, you will also spend an atrocious amount of time on a question clearly designed to reward you for using logic. Let’s look at this question logically and see what we can determine.

Firstly, it only cares about positive integers, so we can disregard zero. This is helpful because a lot of questions hinge on whether or not zero is included, but that won’t matter in this instance. Furthermore, only integers matter, and we’re looking for multiples of 4 but not 6. Your initial pass on a question like this might look might concentrate on the multiples of 4 and you might write (part of) the following sequence down:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100…

After writing a couple of dozen numbers, you may try to figure out the pattern and extrapolate from there. Numbers divisible by 6 are to be eliminated, so you could rewrite this sequence:

4, 8, 16, 20, 28, 32, 40, 44, 52, 56, 64, 68, 76, 80, 88, 92, 100…

Even with this, we have a long sequence of numbers, some of which are crossed off, and less than halfway through the entire sequence. Perhaps approaching the question from a more strategic approach would yield dividends:

The number must be divisible by 4 but not by 6. Calculating the LCM gives us 12, which means that every 12th number will be divisible by both of these numbers. We want the integers to be divisible by four, but not by six, so 12 is out. Along the way, we stop by 4 and 8, both of which are divisible by four but not by six. So every 12 numbers, our count goes up by two, and we start the pattern again. 1-12 will give two numbers that work. 13-24 will give two more numbers that work. 25-36 gives two more, 37-48 gives two more and 49-60 gives two more as well. Thus, through 60 numbers, we have 10 elements that are divisible by 4 and not 6.

From here, it might be easier to go up in bounds of 60, so we know that 61-120 gives 10 more numbers. 121-180 and 181-240 as well. This brings us up to 240 with 40 numbers. A cursory glance at the answer choices should confirm that it must be 42, as all the other choices are very far away. The numbers 244 and 248 will come and complete the list that’s (naughty or nice) under 250. Answer choice C is correct here.

There are other ways to get the right answer, but the fastest ones all hinge on pattern recognition. Figuring out that every 12 numbers gives two more answers can take us from 1 to 240 in one shot (20 sequences x 2). Alternatively, once finding 4 elements at 24, you can probably easily envision multiplying the total by 10 and getting to 240 straight away (like warping over worlds in Super Mario Bros).

Timing is one of the key elements being tested on the GMAT, and one of the goals of the exam is to reward those who have good time management skills. Given 10 minutes, almost everyone would get the correct answer to this question, but the exam wants to determine who can get it right in a fraction of that time. On the GMAT, as in business, timing is everything.
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Re: How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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16 Aug 2015, 05:35
Multiples of 4 are:

4,8,12,16,20,24,28,32,36 - 9 numbers

12,24,36 are multiple of 6 - 3 numbers

so for 62 multiple of 4 we will have 20 multiple of 6

62-20=42

C
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Re: How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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16 Aug 2015, 05:59
Bunuel wrote:
How many positive integers less than 250 are multiple of 4 but NOT multiples of 6?

(A) 20
(B) 31
(C) 42
(D) 53
(E) 64

Kudos for a correct solution.

I solved this differently:
List multiples of 4, making groups of 3:
4,8,12, 16,20,24, 28,32,36...
Every 3rd multiple is a multiple of 6. So in short, a third of all multiples of 4 will be multiple of 6
Total number of multiples of 4: (248-4)/4 +1 = 62.
Now, the 62nd multiple is 248; 61st multiple is 244; 60th multiple is 240 - a multiple of 12
So upto the 60th multiple we will get proper groups of 3.
So 1/3rd of 60 = 20 multiples will be multiple of 6.

So total number of multiples = 62-20 = 42
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Re: How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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23 Aug 2016, 11:05
1
1
In these Kinda Questions its best to use the set theory
Set A => numbers of multiples of 4 => 62
Set B = multiples of 6
We are asked numbers that are only multiples of 4 => Number of elements in set A - number of elements in A intersection B
Hence A intersection B => multiples of 12 => 20
Required Value => 62-20 => 42
SMASH that C
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Re: How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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15 Oct 2016, 21:05
Bunuel wrote:
How many positive integers less than 250 are multiple of 4 but NOT multiples of 6?

(A) 20
(B) 31
(C) 42
(D) 53
(E) 64

Kudos for a correct solution.

Multiples of 4, less than or equal to 250 = [250/4] = 62

Multiples of 6&4 i.e. 12, less than or equal to 250 = [250/12] = 20

Now, Multiples of 4 that are not multiples of 6 = 62-20 = 42

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Re: How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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20 Jan 2017, 04:39
1) The number of multiple of 4 but not multiples of 6=the number of multiples of 4-the number of multiple of 4 AND 6. 2)The number of multiples of 4=(248-4)/4+1=61+1=62 3)The prime factors of 4 are 2*2; the prime factors of 6 are 2*3, thus the prime factors of the numbers that are multiples of 4 AND 6 are 2*2*3=12. 4)The number of multiples of 12=(240-12)/12+1=20 5)62-20=42
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Re: How many positive integers less than 250 are multiple of 4 but NOT mul [#permalink]

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22 Jan 2018, 09:35
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Re: How many positive integers less than 250 are multiple of 4 but NOT mul   [#permalink] 22 Jan 2018, 09:35
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