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  4. The sum of all natural numbers 'n' such that 100 < n < 200 and H.C.F. (91, n) > 1 is :

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Q. The sum of all natural numbers '$n$' such that $100 < n < 200$ and $H.C.F. (91, n) > 1$ is :

JEE MainJEE Main 2019Sequences and Series

Solution:

$S_A$ = sum of numbers between 100 & 200 which are divisible by 7.
$\Rightarrow \; S_A = 105 + 112 + .... + 196$
$S_A = \frac{14}{2} [105 + 196] = 2107$
$S_B$ = Sum of numbers between 100 & 200 which are divisible by 13.
$S_B = 104 + 117 + ... + 195 = \frac{8}{2} [ 104 + 195 ] = 1196$
$S_c$ = Sum of numbers between 100 & 200 which are divisible by both 7 & 13.
$S_c$ = 182
$\Rightarrow $ H.C.F. (91, n) > 1 = $S_A + S_B - S_c = 3121$