Habits of Mind Problem #7
Some Prime Cuts of Problems
Due October 11, 2011
1. A primey is a positive whole number with 3 factors. How many primeys below 2011 are
there?
Since a primey has 3 factors, it must be of the form p2, with p a prime. The factors are 1,
p, and p2 itself. Thus, the question asks how many squares of primes are less than 2011?
Since 442 = 1936 < 2011 and 452 = 2025 > 2011, this is the same as asking how many
primes are less than 44, and the answer is 14, namely
2,3,5,7,11,13,17,19,23,29,31,37,41,43.
2. The ages of a father and his two kids are all different exponents of the same prime
number. A year ago the ages of all three of them were primes. How old are they now?
Consider the ages a year ago, when all were prime. Since all primes except 2 are odd, a
year from now, if the ages are odd primes, the current ages will be even. SO it makes
sense to look for the current ages as powers of 2. Just by inspection, or a little guess-andcheck some reasonable current ages are 22 = 4, 23 = 8 and 25 = 32. The previous ages
were 3, 7, 31 all prime!
3. Find prime numbers x, y, and z, for which 2x+3y+6z=78. Is there some reasoning from
Section 8.4 of our text that you can apply to eliminate some trial-and-error guessing?
If x, y, z were all odd primes, then 2x and 6 z would be even, 3y would be odd, and the sum
would be odd. So choose y = 2 to eliminate this possibility. Then we seek primes x and z to
make 2 x + 6 + 6 z = 78, or 2x + 6z = 72 or x + 3z = 36. One possibility by inspection is x = 3,
and z = 11.
4. There are 67 (white and red) balls in a bowl. There are small ones and big ones among
them. We know that:
a) the number of red balls is divisible by 5;
b) the number of big red balls is the same as the number of white balls;
c) the number of small white balls is the smallest;
d) the number of each kind of balls is a prime.
How many of each of the balls are there in the bowl?
Let R be the number of big Red balls, r the number of small red balls, W the number of
big White balls, and w the number of small white balls. Then we are given that
R + r = 5k for some k
R=W+w
R + r + W + w = 67
and R, r, W and w are primes with w the smallest value. The primes less than 67 are
2,3,5,7,11,13,17,19,23,29,31,37, 41,43,47,53,59, 61. There is no automatic way to solve
this set of equations except through some kind of systematic guess-and-check. Through
that process, I guessed w = 2, and then with educated reasoning r = 53, R = 7, so W = 5.
5. Could the product of the digits of a whole number be 2010?
The prime factorization of 2010 = 2*3*5*67, so the product of the digits must also have
the same factorization, but 67 is prime, so two digits cannot multiply to give this number!
The requested whole number is not possible, NO is the answer.
Statement of Authorship
My (our) signature(s) below indicates that:
1) I (we) did not use any resources such as the web or books other than our textbook.
2) I (we) did not get any help from any individual other than my peers in the class and my instructor.
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3) Below, I (we) have identified any other classmates with whom we discussed this problem
whether they gave us help or we gave them help.
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Individual Responsibility When Group Work is Permitted
Opportunities for group work include both situations when several people sign their name to the same
assignment and when one individual receives help from a classmate even when they do not submit
assignments together. Group work is permitted for Habits of Mind problems, projects, group
presentations, etc. This is done in the belief that, done right, it supports learning, minimizes stress when a
mistake is made, and results in a culture of achievement. Many of us recognize that it is often easier to
work on something that is challenging if we get to share the challenge (and the success) with peers. At the
same time, the learning process can be undermined when one person tells another the solution to a
problem (instead of offering a helpful suggestion) or when someone signs their name to an assignment
when they did not contribute their fair share of effort.
When students work together on a HoM problem, each member of the group should sign their name to
this cover page. Your signature certifies that
i) you have participated to a fair degree in the effort to solve the problem, and thus you are
entitled to your share of the credit for the project;
ii) you understand your group’s solution and will present it if requested; and
iii) everyone you permit to sign the report has also earned the right to sign the solution.