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

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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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So how I approached that question with sequences technique.

1) Total numbers in the set: $$\frac{(99-1)}{1}$$ + 1=99

2) Find # of multiples of 2: $$\frac{(98-2)}{2}$$ +1=49

3) Find # of multiples of 3: $$\frac{(99-3)}{3}$$ +1=33

4) (There is an intersection-overlap between the two above) so find # of multiples of 6: $$\frac{(96-6)}{6}$$ +1=16

5) All the rest: 99 - [(49+33)-16] = 33

To find the respective multiples I narrow the range accordingly by shifting the upper and lower limits to enclose the exact relevant multiples. Say 98 is the largest mulptiple of 2 in the given set from 1 to 99. Though neither 99 nor 98 is a multiple of 6 so I shifted down do 96 at the upper limit and raised the lower limit to the first available multiple of 6 that is 6 itself.

Originally posted by shasadou on 16 May 2016, 09:21.
Last edited by shasadou on 17 May 2016, 04:59, edited 2 times in total.
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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Vyshak
The question states positive integers less than 100. i.e. 1, 2, 3, .... 99

Number of multiples of 2 less than 100 = 49 (2*49 = 98)
Number of multiples of 3 less than 100 = 33 (3*33 = 99)

Some of the integers that are divisible by both 2 and 3 are double counted.
LCM(2, 3) = 6
Number of multiples of 6 less than 100 = 16 (6*16 = 96)

Number of positive integers that are not divisible by 2 or 3 = 100 - (49 + 33 - 16) = 100 - 66 = 34

Hi Vyshak,
Since 100 is being negated as div by 2, it did not make any difference if we take it or not..
But if we are not taking it, as correctly observed by you, DO not take in total--

Quote:
Number of positive integers that are not divisible by 2 or 3 = 100 - (49 + 33 - 16) = 100 - 66 = 34
Here 100 should be 99, and answer will be 99-66 = 33
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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devbond
How many positive integers less than 100 are neither multiples of 2 or 3.

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

Attachment:

Set.png [ 7.64 KiB | Viewed 134629 times ]

Multiples 2 and 3 is ( 50 + 33 ) = 83

Multiple of 6 = 16

While counting the multiples of 2 and 3 we have already included the multiple of 6 ( Which itself is a multiple of 2 & 3 ) so we need to subtract it from the multiples of 2 & 3 to reach the correct answer.

So, the total multiple of 2 and 3 is 67 ( 83 - 16 )

Hence answer will be (D) 67
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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Multiples of 2 = (98 - 0)/2 + 1 = 50
Multiples of 3 = (99 - 0)/3 + 1 = 34
Multiples of 6 = (96 - 0)/6 + 1 = 17

Num of multiples = 50 + 34 - 17 = 67

Numbers that are not multiples of 2 and 3 = 100 - 67 = 33. D

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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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Multiples of 2 = (98 - 0)/2 + 1 = 50
Multiples of 3 = (99 - 0)/3 + 1 = 34
Multiples of 6 = (96 - 0)/6 + 1 = 17

Num of multiples = 50 + 34 - 17 = 67

Numbers that are not multiples of 2 and 3 = 100 - 67 = 33. D

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hi although you reach the right answer your calculation is not accurate and may play bad with you on other questions - that is you will get a wrong answer. The set is exclusive of 0 as the stem clearly limits the numbers to positive ones only. Look at my solution it is slick neat and beautiful lol =). To find the respective multiples I narrow the range accordingly by shifting the upper and lower limits to enclose the exact relevant multiples. Say 100 is a mulptiple of 2 that is why I left it as it is. Though 100 is not the multiple of 6 so I shifted down do 96 at the upper limit and raised the lower limit to the first available multiple of 6 that is 6 itself.
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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Correcting my solution:

Multiples of 2 = (98 - 2)/2 + 1 = 49
Multiples of 3 = (99 - 3)/3 + 1 = 33
Multiples of 6 = (96 - 6)/6 + 1 = 16

Numbers between [1-99] = 99

Num of multiples = 49 + 33 - 16 = 66

Numbers that are not multiples of 2 and 3 = 99 - 66 = 33. D

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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
devbond
How many positive integers less than 100 are neither multiples of 2 or 3.

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

Numbers less than 100 that are neither multiples of 2 or 3.

Multiples of 2 = 2, 4, ... 98 - 49 numbers
Multiples of 3 = 3, 6, 9, ... 99 = 33
Multiples of 6 - 6, 12, ... 96 - 16numbers

Hence total numbers that are either multiples of 2 or 3 = 49 + 33 - 16 = 66
Hence numbers that are not multiples of 2 or 3 = 99 - 66 = 33

Correct Option: D
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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Two ways this question can be approached =>
multiples of 2=>50
Multiples of 3=> 33
Multiples of 6=> 16(BOTH)
either 2 and 3 multiples => 50+33-16=> 77
Neither nor (2,3) => 100-77=> 33

Another approach can be this =>
Non multiples of 2=> 50
multiples of 3 out of these => 3,9,15....99=> 17
hence leftovers => 50-17=> 33

Smash that D
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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stonecold
Two ways this question can be approached =>
multiples of 2=>50
Multiples of 3=> 33
Multiples of 6=> 16(BOTH)
either 2 and 3 multiples => 50+33-16=> 77
Neither nor (2,3) => 100-77=> 33

Another approach can be this =>
Non multiples of 2=> 50
multiples of 3 out of these => 3,9,15....99=> 17
hence leftovers => 50-17=> 33

Smash that D

Bro, although you got the answer correct , you have missed "positive integers less than 100".

So, we need to consider the positive integers from 1 to 99 only.
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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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
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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devbond
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.

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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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devbond
How many positive integers less than 100 are neither multiples of 2 or 3.

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

Multiples of 2: 2, 4, 6, ..., 96, 98
98/2 = 49, so there are 49 multiples of 2

Multiples of 3: 3, 6, 9, ..., 99
99/3 = 33, so there are 33 multiples of 3

At this point we have counted some multiples TWICE. For example, we counted 6 TWICE, we counted 12 TWICE and so on.
In fact, we counted all multiples of 6 TWICE
Multiples of 6: 6, 12, 18..., 96
96/6 = 16, so there are 16 multiples of 6

So.....TOTAL multiples of 2 OR 3 = 49 + 33 - 16 = 66

There are 99 positive integers that are less than 100
So, the TOTAL number of those integers that are NEITHER multiples of 2 or 3 = 99 - 66 = 33

Cheers,
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
chetan2u
devbond
How many positive integers less than 100 are neither multiples of 2 or 3.

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

Hi,

To answer this Q we require to know

1) multiples of 2 till 100 $$= \frac{100}{2} = 50$$
2) Multiples of 3 till 100 = $$\frac{100}{3} = 33.33= 33$$

add the two $$50+33=83$$ ; subtract common terms that are multiple of both 2 and 3..

LCM of 2 and 3 = 6
Multiples of 6 till 100 = $$\frac{100}{6} = 16.66 = 16$$
so total multiples of 2 and 3 = 83-16 = 67

ans = $$100-67 = 33$$

D

When the question says less than 100, why are we including 100 in multiples of 2?
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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TanyaTanya20002
chetan2u
devbond
How many positive integers less than 100 are neither multiples of 2 or 3.

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

Hi,

To answer this Q we require to know

1) multiples of 2 till 100 $$= \frac{100}{2} = 50$$
2) Multiples of 3 till 100 = $$\frac{100}{3} = 33.33= 33$$

add the two $$50+33=83$$ ; subtract common terms that are multiple of both 2 and 3..

LCM of 2 and 3 = 6
Multiples of 6 till 100 = $$\frac{100}{6} = 16.66 = 16$$
so total multiples of 2 and 3 = 83-16 = 67

ans = $$100-67 = 33$$

D

When the question says less than 100, why are we including 100 in multiples of 2?

Hi, you are correct, we don’t require to consider 100.
However, it is not affecting here. Because it is getting cancelled out as multiple of 2.

Say, the question was to find same thing but less than 97. Then, the method would require us to cancel 97 from total because 97 is not divisible by either of 2 and 3, and will be part of the answer.
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
chetan2u
devbond
How many positive integers less than 100 are neither multiples of 2 or 3.

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

To answer this Q we require to know

1) multiples of 2 till 100 $$= \frac{100}{2} = 50$$
2) Multiples of 3 till 100 = $$\frac{100}{3} = 33.33= 33$$

add the two $$50+33=83$$ ; subtract common terms that are multiple of both 2 and 3..

LCM of 2 and 3 = 6
Multiples of 6 till 100 = $$\frac{100}{6} = 16.66 = 16$$
so total multiples of 2 and 3 = 83-16 = 67

ans = $$100-67 = 33$$

D
­Hi chetan Normally we avoid using this approach (biggest number / multiple) as it give and approximate value of the number of multiples. Why does it work here?
I am not good at quant so I may be asking something obvious... Apologies in advance.
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Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
ruis
chetan2u
devbond
How many positive integers less than 100 are neither multiples of 2 or 3.

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

To answer this Q we require to know

1) multiples of 2 till 100 $$= \frac{100}{2} = 50$$
2) Multiples of 3 till 100 = $$\frac{100}{3} = 33.33= 33$$

add the two $$50+33=83$$ ; subtract common terms that are multiple of both 2 and 3..

LCM of 2 and 3 = 6
Multiples of 6 till 100 = $$\frac{100}{6} = 16.66 = 16$$
so total multiples of 2 and 3 = 83-16 = 67

ans = $$100-67 = 33$$

D
­Hi chetan Normally we avoid using this approach (biggest number / multiple) as it give and approximate value of the number of multiples. Why does it work here?
I am not good at quant so I may be asking something obvious... Apologies in advance.
­
There are actually 49 positive multiples of 2, so 49 even numbers less than 100, not 50. Additionally, there are 33 multiples of 3 and 16 multiples of 6. Therefore, to find the number of positive integers less than 100 that are neither multiples of 2 nor 3, we employ the formula: (total) - (multiples of 2) - (multiples of 3) + (multiples of 6) = 99 - 49 - 33 + 16 = 33. However, in the solution, 100 was taken instead of 99, and 50 instead of 49. Fortunately, these balanced each other out, leading to the correct answer.

Nevertheless, I recommend utilizing the conventional method for finding multiples of x within a given range:

• $$\frac{\text{last multiple in the range - first multiple in the range} }{\text{multiple} }+1$$

By employing this method, you'll consistently obtain the exact number without concerns of inaccuracies.­
Re: How many positive integers less than 100 are neither multiples of 2 or [#permalink]
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