shrive555,
Whenever you solve such questions, always think "
How do such numbers look like? What is the form of such numbers?". In this particular question, the answer is:
"13Q + 2" (
since Dividend = Divisor * Quotient + Remainder.)
where "
13" is the divisor, "
2" is the remainder. These two are fixed since these are provided in the question itself & "
Q" is a variable. You can put "
Q" as
any of the integers. I mean any (0, -1, -2 etc). This equation will not be harmed.
Now the question statement says "
How many positive integers ...", Q can not be negative & thus the minimum value that "Q" can have is "0". If we put q ="0" in the equation "13Q + 2", we get 2. Go on & put Q=1 now & you will get 15 and so on. This becomes a series now & it will look like:
2, 15, 28 & so on...
This becomes an arithmetic progression with "13" being the
common difference & "2" being the
first term. In this question, we need to count the number of integers less than 100, so maybe one can jot down such numbers & count them in an exam. These will be:
2, 15, 28, 41, 54, 67, 80, 93 -> count them & answer is 8. It took me not more than 15 seconds to jot them down. I might do the same in an exam.
The bigger question, what if we need to count such numbers which are less than 1000? Can we count them then? The answer is a clear NO. How to proceed then?
2 is nothing but 2 + 13*0 (1st term)
15 is nothing but 2 + 13*1 (2nd term)
28 is nothing 2 + 13*2 (3rd term)
41 is nothing but 2 + 13*3 (4th term)
.
.
.
The nth term will be nothing but
2 + 13*(N-1), correct? (Notice in the 1st term, 13 is getting multiplied with "0". In 2nd term, it is getting multiplied with 1 & so on, in Nth term, it will be getting multiplied with N-1)
Since we need to find integers less than 100, my goal is to find out that Nth term which is less than 100 & follows the same equation.
Thus, my equation becomes:
2 + 13*(N-1)< 100. Solve this & you will get N < (98/13) + 1 i.e. 8.xx (no need to calculate the decimals here).
Thus, the answer is 8.
Try extrapolating the question to less than 1100, 1500, etc. You will become more confident.
Thanks!
_________________
Quant Faculty
CrackVerbal