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1.Set Theory
hard
For an integer $n$ let $S_n=\{n+1, n+2, \ldots \ldots, n+18\}$. Which of the following is true for all $n \geq 10$ ?
A
$S_n$ has a multiple of $19$
B
$S_n$ has a prime
C
$S_n$ has at least four multiples of $5$
D
$S_n$ has at most six primes
(KVPY-2013)
Solution
(d)
We have,
$S_n=\{n+1, n+2, n+3, \ldots, n+18\},$
$n \geq 10$
(a) $n=19$, then $S_n$ has not a multiple of 19 .
Hence, option (a) is false.
(b) $S_n$ has more than one prime for $n \geq 10$.
(c) $n=10$
11,28
Only $15,20,25$ are multiple of $5 .$
(d) Number of odd integer in $S_n=9$
So, every third integer is multiple of $3$.
$\therefore S_n$ has at most six primes.
Standard 11
Mathematics
