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Multiples and integers between

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Multiples and integers between [#permalink] New post 08 Jan 2012, 17:50
I have one concern regarding this How many multiples and how many integers?

How many of the three-digit numbers are divisible by 7?

Generally we do--->999-100= 899

so 899/7=128 so we need to include both digits so we add 1 to total so it is 128+1

but the answer is only 128,How is that possible?????

As per our knowledge we know that


->If the question says both inclusive we add +1 at the last, For eg

How many of the three-digit numbers are divisible by 7?

Generally we do--->999-100= 899

so 899/7=128 so we need to include both digits so we add 1 to total so it is 128+1

->If the question says both not inclusive we add -1 at the last, For eg

How many of the three-digit numbers are divisible by 7?

Generally we do--->999-100= 899

so 899/7=128 so we need to include both digits so we add 1 to total so it is 128-1

->If the question says one inclusive we only subtract two extremes, for Eg

How many of the three-digit numbers are divisible by 7?

Generally we do--->999-100= 899

so 899/7=128, so 128 is the answer.....

Anything wrong with my concept???????Please explain i am totally confused with this...............

If anything wrong,please explain how to with these kind of problems, Like How many integers between two numbers? and How many integers between two multiples?
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Re: Multiples and integers between [#permalink] New post 08 Jan 2012, 22:18
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Hi, there. I'm happy to help with this. :)

First of all, the "add one" thing for inclusive counting only works after subtracting biggest from smallest on a list. If you divide or do some other calculation first, then you can't just add one to that result and call it inclusive counting. You divided by 7, then added one. That's not bonafide inclusive counting.

Next, notice that there are not 899 three digits numbers --- there are actually 900, because you have to count 100. That is genuine inclusive counting --- 999 - 100 + 1 = 900.

Now, for multiples, first find out about the other factors.
7*14 = 98
7*15 = 105, so 7*15 is the first multiple of 7 that's a three-digit number.

1000/7 = 142 and change, so
7*142 = 994
7*143 = 1001, so 7*142 is the last multiple of 7 that's a three-digit number.

How many multiples from 7*15 to 7*142? That's equivalent to: how many integers from 15 to 142? That's where inclusive counting comes in again:

142 - 15 + 1 = 128

Does that make sense? Please let me know if you have any questions about what I've said here.

Here's another similar question for practice.

http://gmat.magoosh.com/questions/846

The question at that link should be followed by a video explanation of the answer.

Mike :)
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Re: Multiples and integers between [#permalink] New post 08 Jan 2012, 23:31
mikemcgarry wrote:
Hi, there. I'm happy to help with this. :)

First of all, the "add one" thing for inclusive counting only works after subtracting biggest from smallest on a list. If you divide or do some other calculation first, then you can't just add one to that result and call it inclusive counting. You divided by 7, then added one. That's not bonafide inclusive counting.

Next, notice that there are not 899 three digits numbers --- there are actually 900, because you have to count 100. That is genuine inclusive counting --- 999 - 100 + 1 = 900.

Now, for multiples, first find out about the other factors.
7*14 = 98
7*15 = 105, so 7*15 is the first multiple of 7 that's a three-digit number.

1000/7 = 142 and change, so
7*142 = 994
7*143 = 1001, so 7*142 is the last multiple of 7 that's a three-digit number.

How many multiples from 7*15 to 7*142? That's equivalent to: how many integers from 15 to 142? That's where inclusive counting comes in again:

142 - 15 + 1 = 128

Does that make sense? Please let me know if you have any questions about what I've said here.

Here's another similar question for practice.

http://gmat.magoosh.com/questions/846

The question at that link should be followed by a video explanation of the answer.

Mike :)


Your explanation is simply superb.......

+1 Kudos for you
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Joined: 28 Jul 2011
Posts: 583
Location: United States
Concentration: International Business, General Management
GPA: 3.86
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Kudos [?]: 40 [0], given: 16

GMAT Tests User
Re: Multiples and integers between [#permalink] New post 08 Jan 2012, 23:41
mikemcgarry wrote:
Hi, there. I'm happy to help with this. :)

First of all, the "add one" thing for inclusive counting only works after subtracting biggest from smallest on a list. If you divide or do some other calculation first, then you can't just add one to that result and call it inclusive counting. You divided by 7, then added one. That's not bonafide inclusive counting.

Next, notice that there are not 899 three digits numbers --- there are actually 900, because you have to count 100. That is genuine inclusive counting --- 999 - 100 + 1 = 900.

Now, for multiples, first find out about the other factors.
7*14 = 98
7*15 = 105, so 7*15 is the first multiple of 7 that's a three-digit number.

1000/7 = 142 and change, so
7*142 = 994
7*143 = 1001, so 7*142 is the last multiple of 7 that's a three-digit number.

How many multiples from 7*15 to 7*142? That's equivalent to: how many integers from 15 to 142? That's where inclusive counting comes in again:

142 - 15 + 1 = 128

Does that make sense? Please let me know if you have any questions about what I've said here.

Here's another similar question for practice.

http://gmat.magoosh.com/questions/846

The question at that link should be followed by a video explanation of the answer.

Mike :)


Can you please refer this link in this link For not inclusive problem "Bunnel"(Gmat user) has added 1 at the last is that correct??....i think there we need to subtract 1 from the count right??

and can you also please throw some light on both not inclusive and 1 inclusive problem...

Thanks in advance
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Re: Multiples and integers between [#permalink] New post 09 Jan 2012, 00:05
how about this:
1st digit: 105
last digit: 994
994=105+(n-1)*7 [AP, nth term]
n=128
Re: Multiples and integers between   [#permalink] 09 Jan 2012, 00:05
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