Math 312, Section 102
Homework #6
due Tuesday, October 30, 2001 at the beginning of class
I. In each of the following numbers, a digit is missing (represented by the letter x). Using
the divisibility tests we’ve learned, find the digit such that the indicated divisibility
is satisfied. Show enough work that I know you didn’t just use trial and error. (All
numbers are written in base 10 unless otherwise specified. Each answer is unique.)
(a) 48144513x is divisible by 16.
(b) 40061178x5 is divisible by 125.
(c) 387x1437 is divisible by 9.
(d) 6033x629 is divisible by 11.
(e) 21423x229 is divisible by 27.
(f) 36x181016 is divisible by 7.
(g) 7x2306831893 is divisible by 9999.
(h) (57329475692x)23 is divisible by 23.
(i) (135x0644)8 is divisible by 7.
(j) (10110100x001001)2 is divisible by 3.
(k) (28x45C0B)16 is divisible by 17.
(l) (6248C96x)15 is divisible by 21. (Hint: check for divisibility by 7 and 3 separately.)
(m) (x032221)5 is divisible by 13. (Hint: 52 + 1 = 13 · 2.)
II. Rosen, Section 5.1, p. 178, #18
III. A British Columbia-wide bowling league is being formed with eight teams respresenting the communities of Alert Bay, Beaverdell, Fernie, Horsefly, Lillooet, Nimpo Lake,
Radium Hot Springs, and Tsawwassen. Once each month, beginning in May, four of
the teams will travel to the other four communities for bowling matches. People are
afraid to visit Radium Hot Springs; consequently, the Radium Hot Springs team will
have to travel every month. Not counting matches Radium Hot Springs, every team
must travel for exactly half their matches. Over the course of the season, each pair of
teams should play exactly once. Schedule the bowling league matches for one season.
(Hint: start by considering the seven teams besides Radium Hot Springs. See Rosen,
Section 5.3, p. 185, #2.)
IV. In the new hit TV show Survivor 4: New Jersey, the format is a little different. The
24 contestants will compete in teams of five through six different contests. The teams
will be shuffled after each contest, so that no two contestants will ever twice be on a
team together. Of course, each contest must have one team of four along with 4 teams
of five; but to be fair, no contestant should be on a four-person team more than once.
Schedule the teams for these contests. (Use A, B, C, . . . , W, X for the contestants and
the labels Contest 1, Contest 2, . . . , Contest 6. Hint: 24 = 52 −1; use the same method
we used in class to schedule the dinners for 9 people, but with the prime 5 everywhere
instead of 3.)