Question

When 13 + 3 ² 1 3 alti
Me mainder
& Find the
is divided by
(1) at​

Answer 1

1+1/2 + 1/3 +….+ 1/23 = n/23!

Well here we have to find remainder when n is divided by 13

As 23! Contains 13 as a factor

Also n = 23! + 23!/2 + 23!/3 +….22!

Also 23! contains 2,3….23 as factors

So we would left with 23!/13 × 13

= (23)(22)…..(14)(12)(11)….2/13

Now we have to find remainder

I prefer to make it in multiple of 13

= (26 -3)( 26–4)…..19…..2/13

From 19 -7 Make it in a unit of 13

= (26–3)(26–4)….(13+6)….(13+1)(13–1)(13–2)…(13–6) . 6.5.4.3.2

Remainder we need , As I have taken 13 and 26 that is multiple of 13

So we left with,

So ( -3)(-4)(-5)(-6)(6)(5)(4)(3)(2)(-1)(-2)(-3)(-4)(-5)(-6).6.5.4.3.2 /13 will give remainder

= (3×4×5×6)4(2)3/13

= (60×6)4×8/13

Make it in unit of 52

= ((52+8)×6)4×8/13

It's remainder now as 52 is multiple of 13

= (8×6)4×8/13

= (48)4×8/13

Make it in unit of 39

=(39+9)4×8/13

As 39 is multiple of 13

So,it's remainder would be same as

Remainder of (9)4×8/13

= (81)2×8/13

Make it in a unit of 78

= (78+3)2×8/13

As 78 is multiple of 13,So it's remainder would be same as

(3)2×8/13

= 72/13

Now 72/13 = (65+7)/13

65ismultipleof13

So , Finally Remainder is 7