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# How many 4-digit positive integers are multiple of each inte

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Intern
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How many 4-digit positive integers are multiple of each inte  [#permalink]

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09 Mar 2010, 07:07
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How many 4-digit positive integers are multiple of each integer from 1 to 10
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10 Mar 2010, 03:06
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lecancher wrote:
Hi,

Can anyone help me with this pls?

how many 4-digit positive integers are multiple of each integer from 1 to 10

4-digit positive integer should be multiple of: 2, 3, 4(=2^2), 5, 6(=2*3), 7, 8(=2^3), 9(=3^2), 10(=2*5). Basically our 4-digit integer should be multiple of LCM of these numbers, which is $$2^3*3^2*5*7=2520$$.

There are 3 such numbers: 2520, 5040, and 7560.
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09 Mar 2010, 11:32
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I am going to assume that you mean 1 to 10 inclusive.

If you wanted to know if a number was divisible by all those digits, you would have to find the smallest # that was divisible by 1 through 10.

1- Every # from 1000 to 9999
1 to 2 - Every even number
1 to 3 - Every multiple of 6 from 1000 to 9999
1 to 4 - Every multiple of 12 from 1000 to 9999
1 to 5 - Every multiple of 60 From 1000 to 9999
1 to 6 - Every multiple of 60 from 1000 to 9999
1 to 7 - Every multiple of 210 from 1000 to 9999
1 to 8 - Every multiple of 840 from 1000 to 9999
1 to 9 - Every multiple of 2520 from 1000 to 9999
1 to 10 - Every multiple of 2520 from 1000 to 9999

With that said, there are three 4-digit numbers that satisfy the requirements of being divisible by 1 through 10 inclusive.
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09 Mar 2010, 12:59
didnt get u.....pls elaborate
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09 Mar 2010, 17:33
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If you mean you didn't get my response. Here is the breakdown:

To get a number that is divisible by all 10 numbers you could do 10!. But 10! = 3628800 and it is not the lower common multiple. To get the LCM, I took all the numbers like this:
1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10.
I removed the 6, 8, 9, 10 and added a 3.
I ended up with 1 X 2 X 3 X 3 X 4 X 5 X 7 which equals 2520
I removed the 6 because it is a multiple of 2 and 3 earlier in the list.
I removed the 8 because it is a multiple of 2 and 4 earlier in the list.
I removed the 10 because it is a multiple of 2 and 5 earlier in the list.
For the 9 to be removed, you needed two 3's. So I added a 3 and removed the 9.

With those adjustments I came up with 2520 as the lowest common multiple for integers from 1 to 10.

Because the question asked between the range of 1000 and 9999, three numbers satisfy the requirements: 2520, 5040, 7560.

Finding the lowest common multiple can be a little confusing. I didn't know how to calculate it so I had to work out how to do this on a smaller subset. I used the one through five range to figure how to figure out the LCM.
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10 Mar 2010, 14:07
gnus wrote:
4-digit positive integer should be multiple of: 2, 3, 4(=2^2), 5, 6(=2*3), 7, 8(=2^3), 9(=3^2), 10(=2*5). Basically our 4-digit integer should be multiple of LCM of these numbers, which is .

There are 3 such numbers: 2520, 5040, and 7560.
lecancher wrote:
Hi,

Can anyone help me with this pls?

how many 4-digit positive integers are multiple of each integer from 1 to 10

4-digit positive integer should be multiple of: 2, 3, 4(=2^2), 5, 6(=2*3), 7, 8(=2^3), 9(=3^2), 10(=2*5). Basically our 4-digit integer should be multiple of LCM of these numbers, which is $$2^3*3^2*5*7=2520$$.

There are 3 such numbers: 2520, 5040, and 7560.

Fabulous approach and solution.

Just goes to show that no matter how much you go when you first read a question that there is almost always a shorter route to the answer, if you are clever enough.
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10 Mar 2010, 19:43
TheSituation wrote:

Fabulous approach and solution.

Just goes to show that no matter how much you go when you first read a question that there is almost always a shorter route to the answer, if you are clever enough.

You nailed the secret to approach GMAT!!! In fact, you are already a Manager. Focus, problem, possibilities and solution, that is all it is in that big bad corporate world. Double Hi Fives bro!!!
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11 Mar 2010, 08:18
BarneyStinson wrote:

You nailed the secret to approach GMAT!!! In fact, you are already a Manager. Focus, problem, possibilities and solution, that is all it is in that big bad corporate world. Double Hi Fives bro!!!

Let's meet up at MacLaren's after work for a pint, we can discuss how we gon run dis town after we takeover.
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09 Aug 2010, 03:56
oldstudent wrote:
Bunuel wrote:
lecancher wrote:
Hi,

Can anyone help me with this pls?

how many 4-digit positive integers are multiple of each integer from 1 to 10

4-digit positive integer should be multiple of: 2, 3, 4(=2^2), 5, 6(=2*3), 7, 8(=2^3), 9(=3^2), 10(=2*5). Basically our 4-digit integer should be multiple of LCM of these numbers, which is $$2^3*3^2*5*7=2520$$.

There are 3 such numbers: 2520, 5040, and 7560.

Hi Bunuel,

Can you please explain how did you get the other 2 nos - 5040 and 7560?

Many thnx

The least 4-digit number which is multiple of each integer from 1 to 10 is LCM of these numbers and equals to $$2^3*3^2*5*7=2520$$.

Now, if we multiply this number by 2 and 3 we will still have 4-digit number which is multiple of each integer from 1 to 10 --> $$2520*2=5040$$ and $$2520*3=7560$$ (if we multiply by 4 the number will be 5-digit). So there are only 3 such numbers.

Hope it's clear.
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09 Aug 2010, 13:51
Bunuel, awesome answer... Can you please check my post on combinations and answer it when you have a chance.. Thanks

Posted from my mobile device
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Re: How many 4-digit positive integers are multiple of each inte  [#permalink]

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05 Mar 2014, 02:18
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Find LCM of nos between 1 & 10
which also means just LCM of 7,8,9 & 10 (As all smaller are multiple of these)

Just 2 is repeated once, else all are unique

7*8*9*5 = 63*40 = 2520

2520, 2520+2520, 2520+2520+2520 is the answer
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Re: How many 4-digit positive integers are multiple of each inte  [#permalink]

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22 May 2016, 07:54
How many 4-digit positive integers are multiple of each integer from 1 to 10

Least No which could be divisible by integers [ 1,2,3,4,5,6,7,8,9,10] is the LCM of the said integers.

LCM [1,2,3,4,5,6,7,8,9,10] = 2520

No the Nos which are divisible by 2520 will be divisible by [1,2,3,4,5,6,7,8,9]

There are 3 values possible between 1000 to 9999

They are 2520 x 1= 2520
2520 x 2 = 5040
2520 x 3 = 7560
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Re: How many 4-digit positive integers are multiple of each inte  [#permalink]

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10 Sep 2018, 09:04
lecancher wrote:
How many 4-digit positive integers are multiple of each integer from 1 to 10

7*8*9*10=5040
5040/2=2520
2520*3=7560
3
Re: How many 4-digit positive integers are multiple of each inte   [#permalink] 10 Sep 2018, 09:04
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