Question

How many numbers lie between $10$ and $300$, which when divided by $4$ leave a remainder $3$?

Answer 1

The number lies between $10$ to $300$, which when divided by $4$ leaves the remainder $3$ are $11,15,19,...299$. Clearly, it is an AP with first term $a=11$, common difference $d=4$ and last term $l=299.$
Let the are n terms in the given AP.
Then, $a_n=299$
$a+(n-1)d=299$
$11+(n-1)4=299\Rightarrow n-1=72\Rightarrow n=73$
Hence, there are $73$ such terms.