In each case show your work and justify your answers. (a) If 9k+5 and 2k+1 have the same last digit for some natural number k, find the last digit of k. (b) If S = ∑ 19 i=1 i!, then find the last digit of S 2 . (c) Denote the last digit of a natural number a by b. Show that the last digit of a n is the same as the last digit of b n where is n ≥ 1 is a natural number. (d) Inspired by part (c), show that the last digit of a 4k+1 for all natural numbers k is the same as the last digit of a. [Euler’s Theorem is not allowed.]

the answer is a i belaeve atleast thats what my calculator just gfave me

step-by-step explanation:

$113.42

i hope that is right.

step-by-step explanation:

answer: the proof is mentioned below.

step-by-step explanation:

since, given:   lines bc = ef

prove:   lines ac > ef

          statements                                           reason

1. bc=ef                                           1. given

2. ac= ab+bc                                 2. because b is the point between the  

                                                            line segment ac

3.   ac > bc                                       3. because if a line segment is equal to

                                                          the sum of two line segments then it is

                                                          always greater then each of these

                                                          individual line segments

4. ac > ef                                         4. by equality property.