Question

Total number of numbers lying in the range of 1331 and 3113 which are neither divisible by 2, 3 or 5 is:

• A. 477
• B. 594
• C. 653
• D. None of these

Answer 1

The correct option is A 477

To find the answer of this problem, first of all, we need to calculate the total number of numbers in the given set which are divisible by any of 2, 3 or 5 then these numbers will be subtracted from the total number of the original set of numbers. Thus we get the total number of numbers which are neither divisible by 2, 3 or 5.
Total numbers in the set = (3113 - 1331) + 1
= 1782 + 1 = 1783
Total number of numbers in the set which are divisible by 2
$=\left(\frac{3112-1332}{2}\right)+1=891$
Total numbers divisible by 3 $=\left(\frac{3111-1332}{3}\right)+1=594$
Total numbers divisible by 5 $=\left(\frac{3110-1335}{5}\right)+1=356$
Total numbers which are divisible by 2 and 3 both
$=\left(\frac{3108-1332}{6}\right)+1=297$
Total numbers which are divisible by 3 and 5 both
$=\left(\frac{3105-1335}{15}\right)+1=119$
Total numbers which are divisible by 2 and 5 both
$=\left(\frac{3110-1340}{10}\right)+1=178$
Total numbers which are divisible by 2, 3 and 5
$=\left(\frac{3090-1350}{30}\right)+1=59$
So, the total number of numbers in the given set which are neither divisible by any of the 2, 3 or 5 = 477.

$\because n(A\cup B\cup C)=\left[n\right(A)+n(B)+n(C\left)\right]-\left[n\right(A\cap B)+n(B\cap C)+n(A\cap C\left)\right]+n(A\cap B\cap C)$
So the divisible numbers
= (891 + 594 + 356) - (297 + 119 + 178) + 59
= 1306
$\therefore$ Number of numbers which are not divisibe by 2, 3 or 5.
= 1783 - 1306 = 477
Hence (a) is the correct option.
Alternatively: Number of numbers in the given set which are only divisible by 2 = 891
the number of numbers which are only divisible by 3, but not by 2.
$=\frac{(3111-1335)}{6}+1=297$
The number of numbers which are only divisible by 5 but not by 2 or 3
$=\{\frac{(3105-1335)}{10}+1\}-\{\frac{(3105-1335)}{30}+1\}$
So the total number which are divisible by any of the 2, 3 or 5 = 891 + 297 + 118 = 1306
Thus the total number of numbers which are neither divisible by 2, 3 or 5 = 1783 - 1306 = 477