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GmaT Practice Exam 6 - Primefactors

This topic has 3 expert replies and 4 member replies

GmaT Practice Exam 6 - Primefactors

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Hello BTG

Got the following question yesterday in the Practice exam 6.
It seem to me, that it is asking for all primefactors less than 100.
But is there a way to quickly calculate this, as I image listing them up and counting is not the best approach.

Thanks in advance
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GMAT/MBA Expert

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Quote:
How many positive integers less than 100 are neither multiples of 2 nor multiples of 3?

a)30
b)31
c)32
d)33
e)34
WRITE IT OUT and LOOK FOR A PATTERN.

1, 2, 3
4, 5, 6
7, 8, 9
10, 11, 12

As illustrated by the values in red, 1 of every 3 positive integers is neither even nor a multiple of 3.
Thus, of the 99 positive integers less than 100, exactly 1/3 will be neither even nor a multiple of 3:
(1/3)(99) = 33.

The correct answer is D.

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Post
Quote:
How many positive integers less than 100 are neither multiples of 2 nor multiples of 3?

a)30
b)31
c)32
d)33
e)34
Another approach:

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

Answer: D

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lucas211 wrote:
Hello BTG

Got the following question yesterday in the Practice exam 6.
It seem to me, that it is asking for all primefactors less than 100.
But is there a way to quickly calculate this, as I image listing them up and counting is not the best approach.

How many positive integers less than 100 are neither multiples of 2 nor multiples of 3?

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

Thanks in advance
I am not sure what you meant by saying "it is asking for all prime factors less than 100"
Are you saying that the question is asking about prime factors of 100??

As for the question, the best way would be to write the numbers down and find a pattern.
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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OptimusPrep wrote:
lucas211 wrote:
Hello BTG

Got the following question yesterday in the Practice exam 6.
It seem to me, that it is asking for all primefactors less than 100.
But is there a way to quickly calculate this, as I image listing them up and counting is not the best approach.

How many positive integers less than 100 are neither multiples of 2 nor multiples of 3?

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

Thanks in advance
I am not sure what you meant by saying "it is asking for all prime factors less than 100"
Are you saying that the question is asking about prime factors of 100??

As for the question, the best way would be to write the numbers down and find a pattern.
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
Hello Optimusprep

Thanks for your reply.
I had misunderstood the question, as I thought it was asking for the prime factors less than the number 100.

Thanks again

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To find the positive integers less than 100 that are not divisible by 2 or 3, we need to find the number of integers that are divisible by 2, 3 and 6 individually.
No of integers divisible by 2 =N2 = 50
No of integers divisible by 3 = N3 = 33
No of integers divisible by 6 = N6 = 16
So, no of integers not divisible by 2 or 3 = 100 - (N2 + N3 - N6) = 100 - (67) = 33.

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To find the positive integers less than 100 that are not divisible by 2 or 3, we need to find the number of integers that are divisible by 2, 3 and 6 individually.
No of integers divisible by 2 =N2 = 50
No of integers divisible by 3 = N3 = 33
No of integers divisible by 6 = N6 = 16
So, no of integers not divisible by 2 or 3 = 100 - (N2 + N3 - N6) = 100 - (67) = 33.

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Quote:
How many positive integers less than 100 are neither multiples of 2 nor multiples of 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 both 2 and 3

Notice that the number of multiples of both 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 numbers from 1 to 99 inclusive that are not multiples of 2 or 3.

Answer: D

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