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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

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22 Feb 2017, 12:11

there are 50 odd nos less than 100 which are not multiples of 2. within these 50 numbers we simply need to remove 50 odd multiples of 3 i.e 3, 9 ,15.......,99. This is an AP series with a=3 d=6 & Tn = 99 Therefore no of odd multiples of 3 less than 100 are: 99 = 3+(n-1)6 = 17 So numbers neither multiple of 2 nor 3 are = 50-17 = 33

How many positive integers less than 100 are neither multiples of 2 or 3.

a)30 b)31 c)32 d)33 e)34

\

We can use the following equation:

Number of integers from 1 to 99 inclusive = number of integers that are multiples of 2 or 3 + number of integers that are neither multiples of 2 nor 3

Furthermore:

Number of integers that are multiples of 2 or 3 = number of multiples of 2 + number of multiples of 3 - number of multiples of 2 and 3

Notice that the number of multiples of 2 and 3 is also the number of multiples of 6.

Let’s determine the number of multiples of 2 from 1 to 99 inclusive using the following equation:

(largest multiple of 2 in the set - smallest multiple of 2 in the set)/2 + 1

(98 - 2)/2 + 1 = 49

Now we can determine the number of multiples of 3 from 1 to 99 inclusive using the same concept:

(99 - 3)/3 + 1 = 33

Finally, let’s determine the number of multiples of 6, since some multiples of 2 are also multiples of 3; we must subtract those out so they are not double-counted.

(96 - 6)/6 + 1 = 16

Thus, there are 49 + 33 - 16 = 66 multiples of 2 or 3 from 1 to 99, inclusive. Therefore, there are 99 - 66 = 33 multiples from 1 to 99 inclusive that are not multiples of 2 or 3.

Answer: D
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]

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15 Apr 2017, 12:44

devbond wrote:

How many positive integers less than 100 are neither multiples of 2 or 3.

a)30 b)31 c)32 d)33 e)34

Set comprises the integers 1-99 inclusive. Number of items in set=99.

Number of integers that are a multiple of 2: [(98-2)/2)]+1=49 Number of integers that are a multiple of 3: [(99-3)/2)]+1=33. Of these, 16 are even and are therefore counted in the number of multiples of 20 (49). So there are 17 additional integers to add that are multiples of 3 but not multiples of 2.

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