The sum of all natural numbers n such that 10 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) We need to find the sum of all natural numbers $n$ such that $100  1$.Since $91 = 7 	imes 13$,the condition $H.C.

Vedclass · Accepted Answer

$3121$

Vedclass · Answer

(D) We need to find the sum of all natural numbers $n$ such that $100  1$.Since $91 = 7 	imes 13$,the condition $H.C.F. (91, n) > 1$ implies that $n$ must be divisible by $7$ or $13$.Let $S_A$ be the sum of numbers between $100$ and $200$ divisible by $7$.The numbers are $105, 112, \dots, 196$.This is an arithmetic progression with $a = 105$,$l = 196$,and $d = 7$.Number of terms $n_A = \frac{196 - 105}{7} + 1 = 14$.$S_A = \frac{14}{2} (105 + 196) = 7 	imes 301 = 2107$.Let $S_B$ be the sum of numbers between $100$ and $200$ divisible by $13$.The numbers are $104, 117, \dots, 195$.This is an arithmetic progression with $a = 104$,$l = 195$,and $d = 13$.Number of terms $n_B = \frac{195 - 104}{13} + 1 = 8$.$S_B = \frac{8}{2} (104 + 195) = 4 	imes 299 = 1196$.Let $S_C$ be the sum of numbers between $100$ and $200$ divisible by both $7$ and $13$ (i.e.,divisible by $91$).The only such number is $182$.So,$S_C = 182$.By the Principle of Inclusion-Exclusion,the required sum is $S = S_A + S_B - S_C$.$S = 2107 + 1196 - 182 = 3121$.

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

Similar Questions

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a given $A.P.$ whose common difference is an integer and $S_{n} = a_{1} + a_{2} + \ldots + a_{n}$. If $a_{1} = 1$,$a_{n} = 300$ and $15 \leq n \leq 50$,then the ordered pair $(S_{n-4}, a_{n-4})$ is equal to

$0.4\overline{23} = $

If $b$ is the first term of an infinite $G.P.$ whose sum is $5$,then $b$ lies in the interval

The sum of $24$ terms of the following series $\sqrt 2 + \sqrt 8 + \sqrt {18} + \sqrt {32} + \dots$ is

Given a sequence of $4$ numbers,the first three of which are in $G.P.$ and the last three are in $A.P.$ with a common difference of $6$. If the first and last terms in this sequence are equal,then the last term is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module