Question

Find sum of all two digit numbers which when divided by 4, yield 1 as remainder

Answer 1

AnswEr:

We have to find the sum of all two digit numbers of the form 4k + 1 , k $\in$ N.

• Clearly, such that numbers are 13, 17, 21, 25, ... , 97 and are forming an A.P. with common difference 4.

Let such numbers be n in number. Then,

________

$\star \: \tiny \bf \: 97 = {n}^{th} \: term \: of \: AP \: with \: first \: term \: 13 \: and \: common \: difference \: 4 \\ \\ \\ \implies \tt97 = 13 + (n - 1) \times 4 \\ \\ \implies \tt \: n - 1 = 21 \\ \\ \implies \tt \blue{ n = 22}$

Let S be the sum of such numbers. Then,

$\\ \qquad \tt \: s = \frac{n}{2}(a _1 + a_n) \\ \\ \\ \qquad \tt \: = \frac{22}{2} (13 + 97) \\ \\ \qquad \tt\orange {s=1210}$