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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 10131
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GMAT 1: 760 Q51 V42
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Among integers from 1 to 50, inclusively, what is the number of the mu
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20 Jan 2020, 00:08
20 Jan 2020, 00:08
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[GMAT math practice question]
Among integers from \(1\) to \(50\), inclusively, what is the number of the multiples of \(4\) or \(5\)?
A. \(14\)
B. \(16\)
C. \(18\)
D. \(20\)
E. \(22\)
Among integers from \(1\) to \(50\), inclusively, what is the number of the multiples of \(4\) or \(5\)?
A. \(14\)
B. \(16\)
C. \(18\)
D. \(20\)
E. \(22\)
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Joined: 03 Oct 2013
Affiliations: CrackVerbal
Posts: 4914
Given Kudos: 221
Location: India
Re: Among integers from 1 to 50, inclusively, what is the number of the mu
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21 Jan 2020, 23:02
21 Jan 2020, 23:02
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This is a very common type of question on the concept of multiples. You can also use the concept of Venn diagrams to solve this question.
In the given interval, the first and the last multiples of 4 are 4 and 48 respectively; so we have 12 multiples of 4.
In the same interval, the first and the last multiples of 5 are 5 and 50 respectively; so we have 10 multiples of 5.
The LCM of 4 and 5 is 20. This means that 20 and its multiples will feature on both of the above lists. Since we are looking at unique representations of each number, we will have to remove the repetitions.
In the given interval, there are 2 multiples of 20 i.e. 20 and 40.
Therefore, number of multiples of 4 or 5 = (Multiples of 4) + (Multiples of 5) – (Multiples of both) = 12 + 10 – 2 = 20.
The correct answer option is D.
Using the concept of Venn diagrams, we can solve the question by drawing a Venn diagram like the one below:
22nd Jan 2020 - Reply 3.jpg [ 42.48 KiB | Viewed 2755 times ]
The circle on the left represents multiples of 4 and has 12 elements; the circle on the right represents multiples of 5 and has 10 elements. The intersection of both circles represents common multiples of 4 AND 5 i.e. LCM of 4 and 5 i.e. multiples of 20; this region has 2 elements.
Therefore, we have 10 elements which are multiples of ONLY 4, 8 elements which are multiples of ONLY 8 and 2 elements which are multiples of BOTH 4 and 5; this is a total of 20 elements.
Hope that helps!
In the given interval, the first and the last multiples of 4 are 4 and 48 respectively; so we have 12 multiples of 4.
In the same interval, the first and the last multiples of 5 are 5 and 50 respectively; so we have 10 multiples of 5.
The LCM of 4 and 5 is 20. This means that 20 and its multiples will feature on both of the above lists. Since we are looking at unique representations of each number, we will have to remove the repetitions.
In the given interval, there are 2 multiples of 20 i.e. 20 and 40.
Therefore, number of multiples of 4 or 5 = (Multiples of 4) + (Multiples of 5) – (Multiples of both) = 12 + 10 – 2 = 20.
The correct answer option is D.
Using the concept of Venn diagrams, we can solve the question by drawing a Venn diagram like the one below:
Attachment:
22nd Jan 2020 - Reply 3.jpg [ 42.48 KiB | Viewed 2755 times ]
The circle on the left represents multiples of 4 and has 12 elements; the circle on the right represents multiples of 5 and has 10 elements. The intersection of both circles represents common multiples of 4 AND 5 i.e. LCM of 4 and 5 i.e. multiples of 20; this region has 2 elements.
Therefore, we have 10 elements which are multiples of ONLY 4, 8 elements which are multiples of ONLY 8 and 2 elements which are multiples of BOTH 4 and 5; this is a total of 20 elements.
Hope that helps!
General Discussion
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 10131
Given Kudos: 4
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Among integers from 1 to 50, inclusively, what is the number of the mu
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22 Jan 2020, 18:19
22 Jan 2020, 18:19
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=>
The number of multiples of \(4\) is \(12 = \frac{(48 – 4)}{4} + 1 = 11 + 1\), since \(4, 8, …, 48\) are the multiples of \(4\) between \(1\) and \(50\), inclusive.
The number of multiples of \(5\) is \(10 = \frac{(50 – 5)}{5} + 1 = 9 + 1\), since \(5, 10, …, 50\) are the multiples of \(5\) between \(1\) and \(50\), inclusive.
Multiples of \(20\) are double-counted, and the number of multiples of \(20\) is \(2\) since \(20\) and \(40\) are multiples of \(20\) between \(1\) and \(50\), inclusive.
Then we have \(12 + 10 - 2 = 20.\)
Therefore, D is the answer.
Answer: D
The number of multiples of \(4\) is \(12 = \frac{(48 – 4)}{4} + 1 = 11 + 1\), since \(4, 8, …, 48\) are the multiples of \(4\) between \(1\) and \(50\), inclusive.
The number of multiples of \(5\) is \(10 = \frac{(50 – 5)}{5} + 1 = 9 + 1\), since \(5, 10, …, 50\) are the multiples of \(5\) between \(1\) and \(50\), inclusive.
Multiples of \(20\) are double-counted, and the number of multiples of \(20\) is \(2\) since \(20\) and \(40\) are multiples of \(20\) between \(1\) and \(50\), inclusive.
Then we have \(12 + 10 - 2 = 20.\)
Therefore, D is the answer.
Answer: D
