How to find number of positive integers ‘x’ such that x/4 has 3 digit & x/3 has 4 digits ?
You might have solved last year CAT papers(when it was available) or any AIMCAT/SIMCAT/..CAT.
A very unique question, isn’t it ?
So what thought process comes in your mind right now ? Probably the very first thing comes in mind is that we should try to check some numbers by dividing 3 and 4 and try to find a pattern.
This happens because many a times we learn that in the tests etc to try problems by using options or putting some values. It of course works for many of the questions but not all questions. So during the test you will have to figure out that in which question you need to apply Conventional approach (DravidPujara approach) & in which question you need to apply Unconventional approach (Sehwag – Dhoni approach).
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But such process takes your 23 minutes of time ans at the end you get no results.
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The reason being you can’t solve such problems until and unless you know the stepwise process to solve it.
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How to find number of positive integers ‘x’ such that x/4 has 3 digit & x/3 has 4 digits ?
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Lets start again with step wise approach.
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Step 1: Think about Range of x w.r.t x/4
E.g. x/4 lies between 100999(as stated in the question). So 400 <= x <= 3996 …………..(i)
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Step 2. Think about range of x w.r.t x/3
For E.g. : x/3 lies between 1000 – 9999 (as stated in the question). So 3000<= x<= 29997………..(ii)
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Step 3: Find the common range of X
For E.g. : From (i) & (ii) it is clear that 3000 <= x <= 3996
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Step 4: Since we know the range of X in which we have to find solution, Now we will check the positive integers X which is divisible by the numbers given in question.
For E.g. : So as per the question, X is divisible by 4 & 3 . i.e. X must be divisible by 12 (LCM of 3 & 4). We will find the number of numbers between 3000 & 3996 which is divisible by 12 and it will give us the answer.
So number of such positive integers = (39963000)/12 +1 = 84.
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Questions for practice :

 Find number of positive integers ‘x’ such that x/5 has 3 digit & x/3 has 4 digits ?

 Find number of positive integers ‘x’ such that x/6 has 4 digit & x/7 has 3 digits ?
 How to find number of positive integers ‘x’ such that x/4 has 4 digit & x/3 has 5 digits ?