Problem Solving

BTGModeratorVI wrote: ↑

Thu Jul 23, 2020 6:48 am

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

A) 30
B) 31
C) 32
D) 33
E) 34

Answer: D

Solution:

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