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

Options

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

Answer

C

Explanation

Clearly, the two digit number which leaves remainder 2 when divided by 7 is of the form `N = 7k + 2` [by Division Algorithm]
For, `k = 2, N = 16`
`k = 3, N = 23`
`k = 13, N = 93`
`:.` 12 such numbers are possible and these numbers forms an AP
Now, `S = (12)/(2) [16 + 93] = 654`
`( :' S_(n) = (n)/(2) (a +l))`
Similarly, the two digit number which leaves remainder 5 when divided by 7 is of the form `N = 7k + 5`
`{:("For",k =1"," N = 12),(,k =2"," N = 19),(,vdots),(,k = 13"," N = 96):}`
`:.` 13 such numbers are possible and these numbers also forms an AP.
Now, `S' = (13)/(2) [12 + 96] = 702 " " ( :' S_(n) = (n)/(2) (a +l))`
Total sum `= S + S' = 654 + 702 = 1356`