Let U be the group of units in Z109 together with the usual multiplication mod 109. Show that U is an abelian group of order 108. (Hint: First note that 109 is a prime number. How many element in U satisfy a2 = 1?)

(a) (8%) compute the probability of an even integer among the 100 integers 1! , 2! , 3! , until 100! (here n! is n factorial or n*(n-1)*(n-2) *… 1) (b) (16%) compute the probability of an even integer among the 100 integers: 1, 1+2, 1+2+3, 1+2+3+4, …., 1+2+3+… + 99, and 1+2+3+… + 100

Labc is reflected across x = 1 and y = -3. what are the coordinates of the reflection image of a after both reflections?

Ok a point t on a segment with endpoints d(1, 4) and f(7, 1) partitions the segment in a 2: 1 ratio. find t. you must show all work to receive credit.