302 PROPERTIES OF THE INTEGERS [CHAP. 1111.87. (a)d = 4 = 5 ( 356 )− 37 ( 48 ); (b)d = 33 =
8 ( 165 )− 1 ( 1287 ); (c)d = 42 = 14 ( 168 )−
1 ( 2310 ); (d)d = 1 = 139 ( 195 )− 28 ( 968 );
(e) 11(249)−74(37).
11.88. (a) 35; (b) 33; (c) 84.
11.89. (a)a= 23 · 3 · 5 · 72 ,b= 22 · 33 · 7 ·11; (b)gcd(a, b)=
22 · 3 ·7, lcm(a, b)= 23 · 33 · 5 · 72 · 11 = 1 , 164 ,240.
11.94. (a) Hint:ak− 1 =(a− 1 )( 1 +a+a^2 +···+ak−^1 );
(b) Hint: Ifn=ab, then 2n− 1 =( 2 a)b−1.
11.98. (k+ 1 )!+2,(k+ 1 )!+3,(k+ 1 )!+ 4 ,...,
(k+ 1 )!+(k+ 1 )are divisible by 2, 3, 4,...,k+1,
respectively.
11.99. (a) False; (b) true; (c) false; (d) false.11.100. (a) 7; (b) 3; (c) 3; (d) 6.
11.l01. (a)−2; (b)−3; (c)−4; (d) 1.11.102. 4, 15, 26, 37, 48, 59, 70, 81, 92.
11.103. −42,−33,−24,−15,−6, 3, 12, 21, 30, 39, 48.
11.104. (a){ 0 , 1 ,..., 10 }and{− 5 ,− 4 ,...,− 1 ,
0 , 1 ,..., 4 , 5 ).
(b){ 0 , 1 ,..., 13 }and{− 6 ,− 5 ,...,− 1 ,
0 , 1 ,..., 6 , 7 }.
11.105.(a){ 1 , 3 }; (b){ 1 , 2 ,..., 10 }; (c) {1, 3, 5, 9,
11,13}; (d) {1, 2, 4, 7, 8, 11, 13, 14}.
11.106. (a) {5, l0, 15, 20, 25, 30, 35,40}; (b) {3, 9, 27, 81,
243, 729, 2187, 6561}.
11.107. m− 1 ≡− 1 (mod m)and so(m− 1 )^2 ≡
1 (mod m).
11.108. (a) 4; (b) 4; (c)=8; (d) 2( 36 ); (e) 4( 55 );
(f)( 23 )( 6 · 75 )( 12 · 132 )
11.109. φ( 8000 ) =φ( 25 · 52 ) = 24 · 4 · 5 =320. Hence
s= 4 ( 320 )=1280.
11.112. (a) 11, 10, 8, 4, 3, 2; (b) 6, 12, 2, 8, 11.
11.113. (a)14,12,11,9,4,2; (b)11,7,3,8,4.
11.114. (a) 17; (b) 61; (c)a−^1 does not exist.
11.115. 2 x^5 + 2 x^2 −x+ 4
11.116. (a) 1, 3, 4; (b) 2,−2; (c) 0, 2, 3,−1.
11.117. (a) 3; (b) 5, 12; (c) no solution.
11.118. (a) 7; (b) no solution; (c) 6, 13, 20.
11.119. (a) 175; (b) 293.
11.120. (a) No solution; (b) 43, 132, 221, 310.
11.121. 49 adults, 7 children.
11.122. 25 apples, 3 pears; 18 apples, 15 pears; 11 apples,
27 pears; or 4 apples, 39 pears.
11.123. (a) 158; (b) 1(123); (c) 3l 415.