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# M16-11

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Math Expert
Joined: 02 Sep 2009
Posts: 46035

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16 Sep 2014, 00:58
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55% (hard)

Question Stats:

63% (01:02) correct 37% (00:53) wrong based on 89 sessions

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How many positive three-digit integers are not divisible by 3 ?

A. 599
B. 600
C. 601
D. 602
E. 603

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Math Expert
Joined: 02 Sep 2009
Posts: 46035

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16 Sep 2014, 00:58
Official Solution:

How many positive three-digit integers are not divisible by 3 ?

A. 599
B. 600
C. 601
D. 602
E. 603

Total 3-digit numbers: $$999-100+1=900$$;

# of multiples of 3 in the given range: $$\frac{\text {last-first}}{multiple}+1=\frac{999-102}{3}+1=300$$;

{Total} - {# of multiples of 3} = {# of not multiples of 3}, hence the answer is $$900-300=600$$.

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Joined: 10 May 2014
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30 Oct 2014, 20:33
2
Hi,

Total 3-digit numbers: 999 - 100 + 1 = 900

We know that the range goes from 100 to 999

There will always be a 2:1 ratio between numbers not divisible by 3 and numbers divisible by 3.
For example: 100, 101, 102... 103, 104, 105... 997, 998, 999

We can infer that the 900 integers in the desired range will also fulfil this 2:1 ratio. Thus, 600 integers are not divisible by 3 and 300 are divisible by 3.

Is my apporach correct?
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Math Expert
Joined: 02 Sep 2009
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31 Oct 2014, 04:58
minwoswoh wrote:
Hi,

Total 3-digit numbers: 999 - 100 + 1 = 900

We know that the range goes from 100 to 999

There will always be a 2:1 ratio between numbers not divisible by 3 and numbers divisible by 3.
For example: 100, 101, 102... 103, 104, 105... 997, 998, 999

We can infer that the 900 integers in the desired range will also fulfil this 2:1 ratio. Thus, 600 integers are not divisible by 3 and 300 are divisible by 3.

Is my apporach correct?

Yes, that's correct.
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13 Oct 2015, 07:21
Hi , the questions says three digit integers why should we not consider -100 to -999 ?

There can be another 600 -ve integers from -100 to -999 which are not divisible by 3.
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18 Oct 2015, 13:47
bissasa wrote:
Hi , the questions says three digit integers why should we not consider -100 to -999 ?

There can be another 600 -ve integers from -100 to -999 which are not divisible by 3.

Edited as suggested. Thank you.
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Joined: 25 Jun 2013
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02 Jun 2016, 20:31
Are you subtracting 102 from 999 because 102 is the first three-digit multiple of 3, and 999 is the last three-digit multiple of 3?
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Joined: 02 Aug 2009
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02 Jun 2016, 20:59
glochou wrote:
Are you subtracting 102 from 999 because 102 is the first three-digit multiple of 3, and 999 is the last three-digit multiple of 3?

Yes this gives us the range of multiples of 3, and in this range every THIRD number will be div by 3....
so we divide the range by 3..
And add 1 as both the first and last integers are div by 3 BUT in our calculation we take ONLY one of them..
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Intern
Joined: 30 Jun 2016
Posts: 3

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11 Jul 2016, 15:01
I did it this way:

We have an arithmetic sequence with 1st term 102 (smallest 3 digit multiple of 3) and 999 (last term), d=3. Using the formula, we have 999 = 102 + 3*(n1) <=> n = 300. Total number of possible 3 digit numbers is 999-100+1 = 900. Deduct those which are divisible by 3 from all possible numbers, 900-300 = 600 are NOT divisible by 3.
Intern
Joined: 05 Oct 2016
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16 Oct 2016, 13:27
Still dont understand why we are adding 1 in the equation. I got 601. Help!
Math Expert
Joined: 02 Sep 2009
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17 Oct 2016, 03:01
nmTL wrote:
Still dont understand why we are adding 1 in the equation. I got 601. Help!

Check this: totally-basic-94862.html

Hope it helps.
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Joined: 02 Sep 2016
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23 Aug 2017, 05:01
Its the same formula as the one for A.P.

Nth term= a+ (n-1)d

Nth term = last term

(Last term- first term)/difference +1= n
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Intern
Joined: 15 Jan 2017
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02 May 2018, 10:04
Hi guys,

I found another way to solve it:

1) There are 999/3 = 333 numbers divisible by 3 [1-999]
2) There are 99/3 = 33 numbers divisible by 3 [1-99]
3) Hence, there are 300 numbers divisible by 3 [100-999]

Since there are 999-100+1=900 numbers (3-digit ), there are 900-300 = 600 positive numbers that are NOT divisible by 3
Re: M16-11   [#permalink] 02 May 2018, 10:04
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# M16-11

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