Question

Find the sum of all such integers less than `100`, which leave a remainder `1` when divided by `3` and leave a remainder `2`, when divided by `4`?

Answer 1

Sum of all such integers less than `100` would be `416`.

Explanation:

Integers which leave a remainder `1` when divided by `3` are

`{4,7,10,13,16,19,22,......}`

and of these, those which leave a remainder `2` when divided by `4` are

`{10,22,34......}`

this is an arithmetic sequence, with first term as `a_1=10` and common difference as `d=12` (note it is LCM of `3` and `4`). As `(100-10)/12=7+....`, the last number in the sequence, up to `100` should be `7+1` or `8^(th)` term.

and sum of the arithmetic sequence given by `S_n=n/2(2a(n-1)d))` would be

`8/2(2xx10+(8-1)xx12)=4(20+84)=4xx104=416`