Question

How many natural numbers between 200 and 400 are there which are divisible by
i Both 4 and 5?
ii 4 or 5 or 8 or 10?

• A. 9, 79
• B. 10, 80
• C. 10, 81
• D. None of these

Answer 1

Answer: (A)

9, 79

The term which are divisible by 8 should also be divisible by 4 and the term which divisible by 10 must be divisible by 5.
So we need to find all the numbers between 200 and 400(excluding 200 and 400) which are divisible 4 and 5.
Now
Total number of natural numbers between 200 and 400 divisible by 4 and 5 = Total number of factors of 4 + Total number of factors of 5-Total number of factors of 20
Now
All numbers between 200 and 400 from an AP with common difference 4 and first term 204 and last term = 396
Let total number of terms be n
Thus $t_n=a+(n-1)d$
$396=204+(n-1)4$
$(n-1)4=192$
$n=49$
Again numbers divisible by 5 from an AP with common difference 5 and first term as 205 and last term as 395
Thus number of term $=\left(\frac{(a_n-a)}{d}\right)+1=\left(\frac{(395-205)}{5}\right)+1=39$
Similarly for numbers of terms divisible by 20 are $=\left(\frac{(380-220)}{20}\right)+1=9$
Thus total number of factors of 4 and 5 $=49+39-9=79$