Question

(h) Find the smallest m such that 2m and 3 (m1) iue system modile 7 composed entirely of (G) Show that n2 - 81n + 1681 is a prime for n 1,2,3,.,80 but not for n = 81. (Note that this problem shows that a sequence of propositions can be valid for many beginning

Answer 1

Solution | gun that 21m and So, we have to find 16+1) the mallest m. lo ороде mal = 3 x m=gry for some NEZ a) for time yer for time put & in me get 254 +1 = 354 zu-254 = 1 (*) now, it in of the Diophantine form antby=( and this equation has a solution if and only if gcd (a,bl divided c. from (*) 623; b=n2 and ged( 35,- 25) = ged(3, 25) = 1 and I divides I hence, it has a solution. Now, by uxing Euclied division Algorithm, we will try to find a andy an fellown.

30-243 = 32x7 +19 an 25-32 2= 32=19x1 +13 19= 13x1+ 6 13 = 6x2 +1 - 6 - lxbt o (ii) fill Now, from (i) we can write IF 13-6x2 - 13 - (19 - 13 ) x2 (from (i)). = 13x3 - 1982 = (3249)* 3 -19x2 (romai)) = 32x3 - 1985 - 32x3 - (243-32x7) X5 (from (iv)) = 32x38 - 243 x5 - 2 x 38 - zx5 1 = 35(-5) - 25(-38) on comparing (kx) from Al we get =-5 and y=-38. Now, other Solution of (*) in given by = -5+(-257 ; 4= -38-357 7621 (Note: an in siophointe equation if we get a solution no. Yo ) then other solution is opuen by Ma not be t ;y=9-a t atta gldab) geacaibl

=-5-25€ y=-38 - 35€ Now, for having smalles positive m. we nowe nyo yo -5-25 130 and 38 - 25t20 and t - 38 = t4 - 00156 and th – 0.156 ant is integer ; t = -1 x and I will be both and smallest one. positive =) na -5+25 = 97 y=- 38+ 3 = 205. from mi = 35 (27) = 6561 => [m> 6560 from @ m. 25 (205) = 6560. is 6560 Hong the ponitive m smallyt