Question

The sum of all two digit positive numbers which when divided by $7$ yield $2$ or $5$ as remainder is :

• A. $1465$
• B. $1256$
• C. $1356$
• D. $1365$

Answer 1

The correct option is C $1356$

Two digit positive numbers that yield $2$ on division with $7$ are of the form $7r+2$ (Where $r\in N$)
Case-1:
$\Rightarrow \sum _{i=2}^{13}(7r+2)=16+23+\cdots +93$
$=\frac{12}{2}[16+93]$
$=654$
Two digit positive numbers that yield $5$ on division with $7$ are of the form $7r+5$ (Where $r\in N$)
Case-2:
$\Rightarrow \sum _{i=1}^{13}(7r+5)=12+19+\cdots +96$
$=\frac{13}{2}[12+96]$
$=702$
Hence, the required sum is $654+702=1356$