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Midterm 1 - Solutions

Midterm 1 - Solutions - 1 Solve the following congruences...

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1. Solve the following congruences. If there are no solutions, say so, and give some justification. (You do not need to give a complete proof.) If there are multiple solutions, be sure to find all of them. (a) 15x+1O=4inZ 20 7 A I (5 6 / 5! 4& ( 15 2J 5 /% 15-jq M t J 2.10 :2Ck 4r ,‘ie k M 5/s 5/2D/ s/ / -21k, Lt 51’l r4 w71*It 12r&l /5’ (oot’ i,)T (b) 15x + 10 = 4 in Z 19 ( I ii) V - /5 33 3- (5q 13 I3(S (d (1) ) 3,(f (3, & ) -(5 d ) / ) i -3 9-(fs-q 4 3)/ = 1’/5$ / \ 1 1= 15 c i (d (T) 7c) 15x +10= 4in 1 -6 (d ) (l f)3 ) 12 (4oL i) e,xfr/y 3 iL’’. 21/) ék VI / 5x9 (Mc 6) d 1) E’’(ii,iJ 6) Szv2 (4lo1) 6 k;
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2. (a) Prove that 10 1 (mod 9) for all n 0. (Hint: Induction) ( ( t J ic’=i& (ioI) Ikt 1 / /./,(q ) 4’ B 1 MI/iM, 1 / () i i (b) Let x be a positive integer, and let y be the sum of the digits of x (in base 10). Prove that x y (mod 9). (Hint: Think about how to write x in terms of its digits. Use part (a).) tilt K -1 - - 2 / 1k z C a, I’ z 4 a 0 a 1 4f+ 414 / fdIi / t dl k (vd) (c) Use part (b) to prove that a positive integer is divisible by 9 if and only if the sums of its digits is divisible by 9. 11 rt ) 4f1V Ifer (K) T UM d i”i (y) i );d/ 1•.
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3. Let R be a ring, and let a E R. Let S={ax I T={xa I xER}. (a)ShowthatforallrERandsES,srES. reJ, i (b) Showthatfora11rERandtT,rtET. / t t 4 Z 6 rt 6 T R / (c) Show
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