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# If 100<= x <=200, then how many Integer values of x are even multiples

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If 100<= x <=200, then how many Integer values of x are even multiples  [#permalink]

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Updated on: 27 May 2017, 10:47
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53% (02:29) correct 47% (02:39) wrong based on 134 sessions

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If $$100 \leq x \leq 200$$, then how many Integer values of x are even multiples of at least one of the numbers 7 and 9?

A) 15
B) 14
C) 13
D) 12
E) 11

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Originally posted by GMATinsight on 27 May 2017, 10:40.
Last edited by Bunuel on 27 May 2017, 10:47, edited 1 time in total.
Formatted the question.
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Re: If 100<= x <=200, then how many Integer values of x are even multiples  [#permalink]

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27 May 2017, 10:56
1
Even multiples of 7 = those which are divisible by both 2 and 7 = those which are divisible by 14 in the range 100 to 200 inclusive
Even multiples of 9 = those which are divisible by both 2 and 9 = those which are divisible by 18 in the range 100 to 200 inclusive

So basically we have to find the numbers of integers from 100 to 200 inclusive which are divisible by at least one of 14 or 18

Multiples of 14 = 112, 126, 140, .... 196 = 7 multiples
Multiples of 18 = 108, 126, 144, .... 198 = 6 multiples

Multiples of both 14 and 18 = Multiples of 126 in the range 100 to 200 inclusive = 1 only (which is 126)

So total numbers in range 100 to 200 inclusive which are divisible by either 14 or 18 or both = 7 + 6 - 1 = 12

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Re: If 100<= x <=200, then how many Integer values of x are even multiples  [#permalink]

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27 May 2017, 15:50
Not sure if my way is correct, but this is how I approached this problem. Between 100 to 200, we can count how many multiples of 7 and 9 there are.

# of multiples of 7 between 100 to 200: 105 to 196- ((196-105)/7)+1)= 14 multiples total
# of multiples of 9 between 100 to 200: 108 to 198- ((198-108)/9)+1)= 11 multiples total

Now, we can count the even multiples. 14 has 7 even multiples and 11 and 5 even multiples. S0, 7+5= 12, Hence, D.

Please let me know if my way is correct.
Thanks!
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If 100<= x <=200, then how many Integer values of x are even multiples  [#permalink]

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10 Jun 2017, 11:08
GMATinsight wrote:
If $$100 \leq x \leq 200$$, then how many Integer values of x are even multiples of at least one of the numbers 7 and 9?

A) 15
B) 14
C) 13
D) 12
E) 11

http://www.GMATinsight.com

The only intuitive way that came to me was to do the times table for 7 & 9 . Starting with 9*1 & 7*1 , then stopping before I hit 100 . Seven has 7 even multiples and nine has 5 even multiples. Since the range was 100 and 200, I knew the number even multiples under 100 would be consistent in the range of 100-200. There is probably some easier equation ,but I try to solve the questions intuitively since I typically cant remember equations I have not practiced in years.
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If 100<= x <=200, then how many Integer values of x are even multiples  [#permalink]

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10 Jun 2017, 14:13
Between 100 and 200,
7 has multiples starting 105(15*7) and ending 196(28*7)
Of these multiples, there are 7 even multiples and 7 odd multiples
9 has multiples starting 108(12*9) and ending 198(22*9)
Of these multiples, there are 6 even multiples and 5 odd multiples

7*9 or 63 has 2 multiples, 126 and 189(one of which is even)

Since we are asked how many integer values of x are even multiples of atleast 7 or 9,
we will have 7+6(13) even multiples of which
126(a number which repeats in both the lists 18*7 and 14*9)
has to be reduced once, making a grand total of 12 even multiples(Option D)
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Re: If 100<= x <=200, then how many Integer values of x are even multiples  [#permalink]

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10 Jun 2017, 19:50
NanaA wrote:
GMATinsight wrote:
If $$100 \leq x \leq 200$$, then how many Integer values of x are even multiples of at least one of the numbers 7 and 9?

A) 15
B) 14
C) 13
D) 12
E) 11

http://www.GMATinsight.com

The only intuitive way that came to me was to do the times table for 7 & 9 . Starting with 9*1 & 7*1 , then stopping before I hit 100 . Seven has 7 even multiples and nine has 5 even multiples. Since the range was 100 and 200, I knew the number even multiples under 100 would be consistent in the range of 100-200. There is probably some easier equation ,but I try to solve the questions intuitively since I typically cant remember equations I have not practiced in years.

The highlighted part seems incorrect. The number of multiple if every range of 100 consecutive numbers is not the same.

e.g. there are 5 even multiples of 9 from 1 to 100
but there are 6 even multiples of 9 from 100 to 200

Word of Caution: Intuition doesn't pay well in GMAT
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Re: If 100<= x <=200, then how many Integer values of x are even multiples  [#permalink]

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10 Jun 2017, 20:03
csaluja wrote:
Not sure if my way is correct, but this is how I approached this problem. Between 100 to 200, we can count how many multiples of 7 and 9 there are.

# of multiples of 7 between 100 to 200: 105 to 196- ((196-105)/7)+1)= 14 multiples total
# of multiples of 9 between 100 to 200: 108 to 198- ((198-108)/9)+1)= 11 multiples total

Now, we can count the even multiples. 14 has 7 even multiples and 11 and 5 even multiples. S0, 7+5= 12, Hence, D.

Please let me know if my way is correct.
Thanks!

Hi ,
You are getting the correct answer but the method may not be entirely correct.
11 multiples does not mean 5 even, it can mean 6 even multiple..
It will depend on the first multiple in that set
Here it's 108 which is even, so 11 will have more even multiples so 6 even and 5 odd..

Total becomes 7+6=13.. but there may be repetition so check for it by taking LCM..
LCM of 7,9 is 63.... There is one even multiple of 63 in the range.. 63*2 as 63*4 will go out of range..
Ans 13-1=12
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Re: If 100<= x <=200, then how many Integer values of x are even multiples   [#permalink] 10 Jun 2017, 20:03
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# If 100<= x <=200, then how many Integer values of x are even multiples

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