Where TRIGONOMETRY meets NUMBER THEORY

Number Theory Level 3

Determine the maximum power of 2 that divides

( 1 + t a n 1 ) ( 1 + t a n 2 ) ( 1 + t a n 3 ) . . . . . ( 1 + t a n 42 ) ( 1 + t a n 43 ) ( 1 + t a n 44 ) (1+tan 1)(1+tan 2 )(1+tan 3).....(1+tan 42)(1+tan 43)(1+tan 44)

DETAILS AND ASSUMPTIONS:

Here x represents x Degrees not x Radians. eg. tan 17 means tan 17 Degrees.

22 This is not an Integer 2 4

Adhiraj Mandal by Adhiraj Mandal , 23, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

( 1 + t a n 1 ) ( 1 + t a n 2 ) ( 1 + t a n 3 ) . . . . ( 1 + t a n 43 ) ( 1 + t a n 44 ) ( 1 + t a n ( 45 − 44 ) ) ( 1 + t a n ( 45 − 43 ) ) . . . ( 1 + t a n 43 ) ( 1 + t a n 44 ) ( 1 + t a n 45 − t a n 44 1 + t a n 45 t a n 44 ) ( 1 + t a n 45 − t a n 43 1 + t a n 45 t a n 43 ) . . . . ( 1 + t a n 43 ) ( 1 + t a n 44 ) S i n c e t a n 45 = 1 , S o , ( 1 + 1 − t a n 44 1 + t a n 44 ) ( 1 + 1 − t a n 43 1 + t a n 43 ) . . . . . ( 1 + t a n 43 ) ( 1 + t a n 44 ) ( 2 1 + t a n 44 ) ( 2 1 + t a n 43 ) ) . . . . . ( 1 + t a n 43 ) ( 1 + t a n 44 ) A l l t h e t e r m s w i l l c a n c e l a n d w e w o u l d b e l e f t w i t h 2 × 2 × 2 × 2 × 2 × 2.....22 t i m e s ( u p t o t a n 22 ) 2 22 S o t h e m a x i m u m p o w e r o f 2 t h a t w o u l d d i v i d e t h i s i s 22 i . e . 2 22 w i l l d i v i d e ( 1 + t a n 1 ) ( 1 + t a n 2 ) . . . . ( 1 + t a n 43 ) ( 1 + t a n 44 ) (1+tan1)(1+tan2)(1+tan3)....(1+tan43)(1+tan44)\\ (1+tan(45-44))(1+tan(45-43))...(1+tan43)(1+tan44)\\ (1+\frac { tan45-tan44 }{ 1+tan45tan44 } )(1+\frac { tan45-tan43 }{ 1+tan45tan43 } )....(1+tan43)(1+tan44)\\ Since\quad tan45=1,\quad So,\\ (1+\frac { 1-tan44 }{ 1+tan44 } )(1+\frac { 1-tan43 }{ 1+tan43 } ).....(1+tan43)(1+tan44)\\ (\frac { 2 }{ 1+tan44 } )(\frac { 2 }{ 1+tan43) } ).....(1+tan43)(1+tan44)\\ All\quad the\quad terms\quad will\quad cancel\quad and\quad we\quad would\quad be\quad left\quad with\\ 2\times 2\times 2\times 2\times 2\times 2.....22\quad times(upto\quad tan22)\\ { 2 }^{ 22 }\\ So\quad the\quad maximum\quad power\quad of\quad 2\quad that\quad would\quad divide\quad this\quad is\\ 22\quad i.e.\quad { 2 }^{ 22 }\quad will\quad divide\quad (1+tan1)(1+tan2)....(1+tan43)(1+tan44)

CHEERS!!!

3 Helpful 0 Interesting 0 Brilliant 0 Confused

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...