Download 201109 Math 122 Assignment 4

201109 Math 122 Assignment 4
Due: Wednesday, November 16, 2011 at the start of class
The marks available are as shown. Please give clear and coherent solutions, showing all steps.
Solutions that are complete and correct will receive full marks. Solutions that are complete and
substantially correct will receive about 2/3 of the marks available. Incomplete solutions will receive
at most 1/3 of the marks available. Minor deductions will be made for small “typographical” errors.
1. (10 marks; 5 per part) Page 97, #20.
2. (a) (3 marks) Page 145, #1.
(b) (2 marks) What happens if the large number in (a) has even more 9 digits?
3. (a) (5 marks) In base ten, a number (dn dn−1 . . . d1 d0 )10 is even if and only if its ones digit,
d0 , represents an even number. Show that this statement is still true when ten is replaced
by any even base b = 2k, k ∈ N.
(b) (6 marks) Show that, in an odd base b = 2k + 1, k ∈ N , a number (dn dn−1 . . . d1 d0 )b
is even if and only if an even number of its digits represent odd numbers. (Hint: this is
the same as the sum of its digits di being even.)
4. Suppose a and b are integers with ab = −27 38 52 76 and gcd (a, b) = 23 34 5.
(a) (5 marks) Is it possible for a to equal 25 34 5? Why or why not?
(b) (3 marks) What is the least common multiple lcm(a, b)?
5. (6 marks; 3 per part) Page 146, #19.
6. (Bonus Question; 5 marks) On an infinite sheet of graph paper, every square cell has a positive
integer. Furthermore, every cell’s number is the average of the 4 surrounding cells’ numbers.
Prove that every cell contains the same number. Hint: consider the smallest number in the
grid (why does one exist?), and use contradiction.