MATH4455
Mathematical Problem Solving for Secondary
Teachers
Problem Set 4
Due Wednesday, March 1 (changed from February 27)
As usual, show your work, and groups are encouraged!
Exercise 1.
(a) Which numbers have exactly 3 divisors? Which numbers have exactly 5
divisors? For an arbitrary given prime q, which numbers n have exactly
q divisors (that is, N D(n) = q) ?
(b) Let M > 1 be composite (that is, not prime), say M = cd where c, d > 1.
(i) Let r be any given prime number. Find a power of r which has
exactly M divisors.
(ii) Find a number n which is not a prime power, but also has exactly
M divisors.
For part (b), answer in terms of M and r for (i), and in terms of
M, c, and/or d for (ii).
(c) Find the smallest positive integer which has exactly M = pq divisors,
where p and q are primes with p < q . No proof required, but explain
why you think it is the smallest one. (Answer in terms of p and q.)
Exercise 2. Explain why the digit jumble trick works.
Exercise 3. Suppose that p is a prime which can be written in the form
p = 2q − 1 for some integer q.
For example, p = 3 and q = 2 or p = 7 and q = 3 are of this form.
Note that p has to be odd. Now let n = 2q−1 p.
(a) Find N D(n) in terms of p and q.
(b) Find SD(n) in terms of p and q.
(c) Find SD(n) in terms of n (not having p and q in your final formula) Hint:
Take your answer to (b), then simplify, simplify, simplify.
Exercise 4. Variation on A Thousand Points of Light
A thousand simple on-off lamps are lined up and are initially off. Person 1
flips every switch once. Person 2 flips every second switch 2 times. Person j
flips every jth switch j times. The last one flips switch 1000 one thousand
times. Which lamps are on when it is over? Give a precise description of this
set of lamps as simply as you can(*).
(*) Degrees of Simplicity of your Final Description:
• Just OK: Using the word divisor in your final description.
• A bit better: Referring to prime factors or exponents in your final answer,
but not divisor.
• Best: Not using any of the words divisor, prime, factor, power, or exponent .
Exercise 5.
(a) For n = 1, 2, ..., 10, make a table of the values of (i) M (n), and (ii) N D(n2 ),
where M (n) is the total number of integer solutions, to xy = n(x + y)
(Module 2, How Many Solutions and HW problems #2 and #3; look at
the solutions to HW 2 #3.)
For example M (1) = 2, M (2) = 6, M (4) = 10, M (5) = 6, ....
(b) Look at your table in (a). Suggest a formula for M (n) in terms of the ND
function and n.
(c) Let n and x be integers and let j = x − n. Show that j divides n2 if and
only if j divides nx.
(d) Using part (c), explain why the formula for M (n) in part (b) is correct.
Hint for (c): Looking back at Module 2, M (n) equals the number of
(positive and negative) integers x such that (x − n) divides nx. Carry on
....
This problem completes the computation of the number of integer solutions
to xy = n(x + y) for arbitrary positive n.
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