Question

How many composite numbers are there in between \[150\] and $200$ (including \[150\] and $200$ both) ?
A) $40$
B) $42$
C) $41$
D) $39$
E) None of these

Answer 1

Hint: There are fifty one numbers in total. Among them we can separate composite numbers by taking multiples of two, three, five, seven, eleven and thirteen. Thus we get the count of composite numbers.

Complete step-by-step answer:
Composite numbers are those numbers which have factors other than one and the number itself.
Numbers which are not composite are called prime.
There are a total of $200 - 149 = 51$ numbers between $150$ and $200$ which includes both the numbers.
Every even numbers among this are composite since they have the factor two.
They all count to $26$ in number.
Now we have $51 - 26 = 25$ numbers remaining.
They are,
$
  151,153,155,157,159 \\
  161,163,165,167,169 \\
  171,173,175,177,179 \\
  181,183,185,187,189 \\
  191,193,195,197,199 \\
 $
Let us find the composite numbers among them.
We have,
$153,159,165,171,177,183,189,195$ are all multiples of three. So they are all composite.
They count to $8$.
Now we have remaining numbers,
$151,155,157,161,163,167,169,173,175,179,181,185,187,191,193,197,199$
Among them,
$155,175,185$ are multiples of five.
$161$ is multiples of seven.
$187$ is a multiple of $11$.
$169$ is a multiple of $13$.
So, the total number of composite numbers is equal to $26 + 8 + 3 + 1 + 1 + 1 = 40$.
Therefore the answer is option A.


Note: Here we considered multiples of two, three, five, seven, eleven and thirteen. We choose these since they are prime numbers. Remaining numbers need not be considered because they will be multiples of any of these numbers. Next prime number is seventeen. Multiples of $17$, in this period are $153,170$ and $187$ which are already counted as multiples of other primes.
And we have $151,157,163,167,173,179,181,191,193,197,199$ are the prime numbers.