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# The integers from 1 to 100 inclusive are each written on a

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Manager
Joined: 15 Jul 2004
Posts: 68
The integers from 1 to 100 inclusive are each written on a  [#permalink]

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09 Apr 2005, 00:13
3
12
00:00

Difficulty:

65% (hard)

Question Stats:

62% (01:58) correct 38% (01:53) wrong based on 787 sessions

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The integers from 1 to 100 inclusive are each written on a single slip of paper and dropped into a jar. If one slip of paper is removed at random, approximately what is the probability that the number on it is neither even nor a multiple of 3?

A. 83%
B. 67%
C. 50%
D. 33%
E. 17%

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Re: The integers from 1 to 100 inclusive are each written on a  [#permalink]

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19 Mar 2012, 02:48
2
4
Zem wrote:
The integers from 1 to 100 inclusive are each written on a single slip of paper and dropped into a jar. If one slip of paper is removed at random, approximately what is the probability that the number on it is neither even nor a multiple of 3?

A. 83%
B. 67%
C. 50%
D. 33%
E. 17%

# of multiples of 2 in the range (100-2)/2+1=50 (check this: totally-basic-94862.html#p730075);
# of multiples of 3 in the range (99-3)/3+1=33;
# of multiples of both 2 and 3, so multiples of 6, in the range (96-6)/6+1=16 (to get the overlap of above two sets);

Hence there are total of 50+33-16=67 numbers from 1 to 100, inclusive, which are multiples of 2 or 3, which means that the probability of selecting neither multiple of 2 nor multiple of 3 is 1-67/100=33/100=33%.

Hope it's clear.
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VP
Joined: 30 Sep 2004
Posts: 1415
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09 Apr 2005, 00:44
1
67%...

50/100 (all multiples of 2) + 33/100 (all multiples of 3) - 1650/10000 (multiples of 3 and 2) =6650/10000=67%...
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Joined: 19 Feb 2005
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09 Apr 2005, 01:39
1
odd multiples of 3 are 17/100
3 3+6 3+6+6 3+6+6+6 etc
50/100+17/100=67/100
Senior Manager
Joined: 15 Mar 2005
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09 Apr 2005, 02:20
2
1
Nither even nor a multiple of 3.

Even numbers between 1 and 100 are 2, 4, 6 ... 100 (50 numbers).

Numbers that are multiple of 3 are 3, 6, 9 ... 99. However, all even multiples of 3 have been counted already in even numbers. Therefore, this set includes 3, 9, 15, 21 ... 99. (17 numbers).

Thus total numbers that are to be excluded = 50 + 17 = 67.

Therefore probability that a randomly picked number is nither even nor a multiple of 3 = 33%
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09 Apr 2005, 02:23

Options >=50% should be ruled out soon because numbers range from 1 to 100, and thus subtracting 50 even you can't have more than 49 (you know at least that 3 is an odd multiple of 3, son you have at least 1 in 50 remaining numbers)

This question is asking nothing more than:
"how many odd numbers are in the first 100 positive ones that are not multiple of 3?"
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10 Apr 2005, 02:37
thearch wrote:

Options >=50% should be ruled out soon because numbers range from 1 to 100, and thus subtracting 50 even you can't have more than 49 (you know at least that 3 is an odd multiple of 3, son you have at least 1 in 50 remaining numbers)

This question is asking nothing more than:
"how many odd numbers are in the first 100 positive ones that are not multiple of 3?"

Careful theArch, that wall is rather fragile !!!

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10 Apr 2005, 02:46
1
However, I was deceived by christoph, because I calculated 67/100 and then I looked at answers already given, and forgot to subtract. This wouldn't happen in the actual test
Manager
Joined: 17 Dec 2004
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10 Apr 2005, 05:49
2
33%

100 slips of paper: 50 are even, and 17 are odd multiples of 3

100-(50+17)=33
Director
Joined: 18 Feb 2005
Posts: 635

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10 Apr 2005, 05:59
thearch wrote:

Options >=50% should be ruled out soon because numbers range from 1 to 100, and thus subtracting 50 even you can't have more than 49 (you know at least that 3 is an odd multiple of 3, son you have at least 1 in 50 remaining numbers)

This question is asking nothing more than:
"how many odd numbers are in the first 100 positive ones that are not multiple of 3?"

This happens with many of us....We get there to the answer and forget subtracting.......GMAT has both the answers as a trap and we should be careful enough not to fall in it........
Senior Manager
Joined: 15 Mar 2005
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11 Apr 2005, 11:24
Zem wrote:
but how you found the number: 17?

Zem, we are trying to find the numbers that are nither divisible by 2 nor 3.

If you take the numbers that are divisible by 2 away, you've taken all even numbers between 1 and 100 away. They are (100-1)/2 = 50 numbers.

We could repeat the same process with 3. However, there are numbers that are common to 2 and 3, and we don't have to take them out (they were accounted for in the even numbers list anyway).

So we see what are these numbers we're looking at (odd multiples of 3).
They are 3, 9, 15, 21 and so on.

There're a plathora of ways to figuring out how many such numbers would exist between 1 and 100. I'd list out two here.

One is the way of sequences.

the last number in this list would be 99.
99 = 3 + (n - 1) 6 => n = 17.

Else do a 100/16 = 16.xxx. 16*6 = 96. but since it starts at 3, the 16th multiple would be 99. 16 multiples, and add one (for 3) and you've 17.

Thus we have the number 17.

Hope that helps.
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Joined: 13 Jun 2004
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11 Apr 2005, 19:30
I think that in this problem it's easier to find directly the prob than to use the formula 1-(opposite prob)

from 1 to 100 : 100 numbers
even numbers : 50
multiples of 3 : 33 (3*1, 3*2, ...3*33) however in those 33 numbers there are already even numbers and you have to be careful to not repeat the even numbers you've already taken out so finally you will get 17 odd and 16 even (you can see that everytime 3 is multiply by an even number the final result is obviously even so 1/2 of the multiples of 3 will be even and 1/2 odd, howver the last one is odd so there is an additionnal odd one)

17 odd multiples of 3 + 50 even numbers = 67

100-67 = 33
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Re: The integers from 1 to 100 inclusive are each written on a  [#permalink]

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18 Mar 2012, 19:56
1
Simple to solve using P(n) = 1-X concept

P = 1 - [ P(even 0r multiples of 2)+P(multiples of 3) - P(multiples of 2 and 3 or multiples of 6)]
P = (1 - [ 50/100+33/100 - 16/100]) = 33%
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Re: The integers from 1 to 100 inclusive are each written on a  [#permalink]

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10 Dec 2013, 05:19
Zem wrote:
The integers from 1 to 100 inclusive are each written on a single slip of paper and dropped into a jar. If one slip of paper is removed at random, approximately what is the probability that the number on it is neither even nor a multiple of 3?

A. 83%
B. 67%
C. 50%
D. 33%
E. 17%

Bunuel is this approach correct?

Ballparking from 1 to 10 there are only 3 numbers that satisfy these conditions: 1,5 and 7. Then 3/10 so approximately 30% only answer choice that is close is D

If you do it for other ranges you are basically going to have 4,3,4,3,4,3 so in theory it would be something like 3.5/10 or between 30% and 35%

Cheers
J
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Re: The integers from 1 to 100 inclusive are each written on a  [#permalink]

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10 Dec 2013, 05:22
Zem wrote:
The integers from 1 to 100 inclusive are each written on a single slip of paper and dropped into a jar. If one slip of paper is removed at random, approximately what is the probability that the number on it is neither even nor a multiple of 3?

A. 83%
B. 67%
C. 50%
D. 33%
E. 17%

Multiple of 3: 100/3 = 33 numbers
Multiple of 2: 100/2 = 50 numbers

Multiple of 6: 100/6= 16 numbers

Total numbers divisible by 2 or 3: 50 + 33 - 16 = 67

Neither = 100 - 67 = 33

Probability = 33/100

Pushpinder Gill
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Re: The integers from 1 to 100 inclusive are each written on a  [#permalink]

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03 Jan 2016, 08:46
Even numbers = 100/2 = 50
Numbers divisible by 3 = 99/3 = 33
Even or divisible by 3 =
50 + 33 - duplicates(divisible by 3 and even)
Duplicates = 32/2 = 16
Even or divisible by 3 = 50 + 33 - 16
= 67
Numbers that are not even or divisible by 3 = 100 - 67 = 33
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Re: The integers from 1 to 100 inclusive are each written on a  [#permalink]

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14 Mar 2016, 00:12
Even => 50
Multiples of 3 => 33
multiples of 6 => 16
Not cases => 67
cases =33
P(cases) = 33/100
percentage =33
hence D
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Re: The integers from 1 to 100 inclusive are each written on a  [#permalink]

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20 Oct 2018, 04:06
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Re: The integers from 1 to 100 inclusive are each written on a   [#permalink] 20 Oct 2018, 04:06
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