Question

The sum of all two digit numbers which are not divisible by $2$ or $3$ is

• A. $1734$
• B. $1620$
• C. $1482$
• D. none of these

Answer 1

Answer: (B)

$1620$

Sum of all digits from 10 to 99 is $S_n=\frac{n}{2}[2a+(n-1\left)d\right]S_n=\frac{90}{2}\left[2\right(10)+(90-1\left)1\right]S_n=4905$.
Sum of digits divisible by 2 from 10 to 99 is $S_n=\frac{n}{2}[2a+(n-1\left)d\right]S_n=\frac{45}{2}\left[2\right(10)+(45-1\left)2\right]S_n=2430$
Sum of digits divisible by 3 from 10 to 99 is $S_n=\frac{n}{2}[2a+(n-1\left)d\right]S_n=\frac{30}{2}\left[2\right(12)+(30-1\left)3\right]S_n=1665$
Sum of digits divisible by 6 from 10 to 99 is $S_n=\frac{n}{2}[2a+(n-1\left)d\right]S_n=\frac{15}{2}\left[2\right(12)+(15-1\left)6\right]S_n=810$
The sum of all two digit numbers which are not divisible by $2$ or $3$ is $4905-2430-1665+810$ is $1620.$