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Manager  B
Joined: 08 Jul 2017
Posts: 88
GMAT 1: 620 Q48 V29
How many positive integers less than 20 are either a multiple of 2,  [#permalink]

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4 00:00

Difficulty:   75% (hard)

Question Stats: 44% (01:52) correct 56% (02:05) wrong based on 62 sessions

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How many positive integers less than 20 are either a multiple of 2, an odd multiple of 7, or the sum of a positive multiple of 2 and a positive multiple of 7 ?

(A) 19
(B) 18
(C) 17
(D) 16
(E) 15

Originally posted by robu1 on 29 Jul 2019, 21:21.
Last edited by Bunuel on 29 Jul 2019, 22:01, edited 1 time in total.
Edited the tags.
Manager  B
Joined: 02 Jun 2014
Posts: 65
Location: India
GMAT 1: 720 Q49 V40
Re: How many positive integers less than 20 are either a multiple of 2,  [#permalink]

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Multiples of 2 -> 9
( 2 4 6 8 10 12 14 16 18 )

Odd multiple of 7 -> 1
( 7 )

Sum of 2 and 7 -> 8
( 7x1 + n*2 = 9 11 13 15 17 19 )

( 7x2 + n*2 = 16 18 )

What am i missing ?

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Manager  B
Joined: 02 Jun 2014
Posts: 65
Location: India
GMAT 1: 720 Q49 V40
Re: How many positive integers less than 20 are either a multiple of 2,  [#permalink]

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Why isnt the answer 18 ?

Posted from my mobile device
Math Expert V
Joined: 02 Sep 2009
Posts: 64314
Re: How many positive integers less than 20 are either a multiple of 2,  [#permalink]

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chetansood wrote:
Multiples of 2 -> 9
( 2 4 6 8 10 12 14 16 18)

Odd multiple of 7 -> 1
( 7 )

Sum of 2 and 7 -> 8
( 7x1 + n*2 = 9 11 13 15 17 19 )

( 7x2 + n*2 = 16 18)

What am i missing ?

Posted from my mobile device

16 and 18 are counted twice.

This question is a copy of the following OG question: https://gmatclub.com/forum/how-many-pos ... 71108.html

Other similar questions to practice:
http://gmatclub.com/forum/how-many-posi ... 06559.html
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https://gmatclub.com/forum/how-many-pos ... 15992.html
https://gmatclub.com/forum/how-many-pos ... 94190.html
https://gmatclub.com/forum/how-many-pos ... 44535.html
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Re: How many positive integers less than 20 are either a multiple of 2,  [#permalink]

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robu1 wrote:
How many positive integers less than 20 are either a multiple of 2, an odd multiple of 7, or the sum of a positive multiple of 2 and a positive multiple of 7 ?

(A) 19
(B) 18
(C) 17
(D) 16
(E) 15

How many positive integers less than 20 are either a multiple of 2, an odd multiple of 7, or the sum of a positive multiple of 2 and a positive multiple of 7 ?
multiple of 2 covers all even numbers from 2-18
sum of a positive multiple of 2 and a positive multiple of 7 covers all odd numbers from 7-19
=> numbers 1,3,5 are the only numbers left out.
=> number of positive integers less than 20 = 19-3 = 16

IMO D
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Re: How many positive integers less than 20 are either a multiple of 2,  [#permalink]

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robu1 wrote:
How many positive integers less than 20 are either a multiple of 2, an odd multiple of 7, or the sum of a positive multiple of 2 and a positive multiple of 7 ?

(A) 19
(B) 18
(C) 17
(D) 16
(E) 15

need to find +ve integers which are less than 20 are
either a multiple of 2, ; ( 2 to 18 ) total 9
odd multiple of 7,; [; 1
or the sum of a positive multiple of 2 and a positive multiple of 7 ; ( 2,7) ( 4,7), ( 6,7) ( 8,7)(10,7) ( 12,7) ; 6
sum of even multiple of 7 and multiple would be an overlap so 18 is not possible i.e ( 2,14)(4,14) chetansood
so total integers ; 9+1+6 ; 16
IMO D
Manager  G
Joined: 08 Jan 2018
Posts: 90
Re: How many positive integers less than 20 are either a multiple of 2,  [#permalink]

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Positive integers less than 20: So we need to consider numbers from 1 to 19 inclusive.
Are either a multiple of 2: 9 numbers will be multiple of 2. {2, 4, 6, 8, 10, 12, 14, 16, 18} -> (a)
OR an odd multiple of 7: Only 1 odd multiple of 7 will be in the given range. {7} -> (b)

OR the sum of a positive multiple of 2 and a positive multiple of 7: Multiple numbers qualify in the given range. We need to make sure we that we do not double count any "even" value as they will be covered in (a)
Sum of a positive multiple of 2 and a positive multiple of 7: (2 + 7 = 9), (2 + 14 = even), (4 + 7 = 11), (6 + 7 = 13), (8 + 7 = 15), (10 + 7 = 17), (12 + 7 = 19), (4 + 14 = even) - So 8 number out of which 2 are already present in (a). Thus, we count 6 here. -> (c)

(a) + (b) + (c) = 16

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Re: How many positive integers less than 20 are either a multiple of 2,  [#permalink]

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robu1 wrote:
How many positive integers less than 20 are either a multiple of 2, an odd multiple of 7, or the sum of a positive multiple of 2 and a positive multiple of 7 ?

(A) 19
(B) 18
(C) 17
(D) 16
(E) 15

The multiples of 2 are:

2, 4, 6, 8, 10, 12, 14, 16, 18

The only odd multiple of 7 less than 20 is 7.

The sums of multiples of 2 and multiples of 7 are:

9, 11, 13 ,15 ,16, 17, 18, 19

However since 16 and 18 are already counted, there are 9 + 1 + (8 - 2) = 16 numbers that fit the criteria.

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If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Re: How many positive integers less than 20 are either a multiple of 2,   [#permalink] 05 Aug 2019, 18:16

# How many positive integers less than 20 are either a multiple of 2,  