TIFR CSE 2010 | Part A | Question: 20

Question

How many integers from $1$ to $1000$ are divisible by $30$ but not by $16$?

• A. $29$
• B. $31$
• C. $32$
• D. $33$
• E. $25$

Answer 1

Option A. i.e., $29$ should be the answer.

Number of integers between $1$ to $1000$ that are divisible by $30 = \lfloor\frac{ 1000}{30}\rfloor = 33.$

Now, since LCM of $30,16 = 240$, only the numbers that are divisible by $240$ between $1$ to $1000$ will divisible by both $30$ & $16.$
So, number of integers between $1$ to $1000$ that are divisible by $240$ $($i.e., divisible by both $30$ and $16 = \lfloor \frac{1000}{240}\rfloor = 4.$

So, total number of integers that are divisible by $30$ but not divisible by $16 = 33 - 4 = 29.$

Answer 2

A: TOTAL NUMBERS DIVISIBLE BY 30

B: TOTAL NUMBERS DIVISIBLE BY 16

P(A-B) = P(A) - P(A∩B) (DIVISIBLE BY 30 BUT NOT BY 16)

           = 33 - 4=29 OPTION A