{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Midterm 1 - Solutions

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

This preview shows pages 1–4. Sign up to view the full content.

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;

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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•.
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern