dan's math@home - problem of the week - archives
Problem Archives page 1

1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100
101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151+ . prob index

1- The Most Divisors
2 - Trains versus Fly!
3 - Unit fractions 1/n
4 - The Census Taker
5 - Square root probs
7 - Name that pattern!
9 - Strange Powers !!
10- Odd & Abundant?

Problem #1 - Posted Friday, November 21, 1997
A "divisor" of a positive whole number n is a whole number that divides evenly into n
(with no remainder).What integer from 1 through 1000 has the most divisors?
To win you must prove it and explain your method.

Solution:
A number with a lot of small prime factors will have lots of divisors.
For example, 12 = 2 * 2 * 3 and 12 has 6 divisors: 1, 2, 3, 4, 6, and 12.
Since 12 has 6 divisors, we say d(12) = 6, where d is the "divisor function."
Restricting to 2's and 3's, 2^5 * 3^3 = 864 and d(864) = 24
Testing out numbers like n = 2^a * 3^b * 5^c we find that
2^4 * 3^2 * 5 = 720 and d(720) = 30, also
2^6 * 3 * 5 = 960 and d(960) = 28, but if we allow 7's then
2^3 * 3 * 5 * 7 = 840 and d(840) = 32.
This is the answer: 840 has 32 divisors.

I call it "supercomposite" because it has more divisors than any smaller number.
The first few supercomposites are:
2 (2 divisors), 4 (3), 6 (4), 12 (6), 24 (8), 36 (9), 48 (10), and 60 (12).

There is a way to find d(n) from the prime factorization of n; it has to do with the exponents, not the primes, so it pays to use higher powers of smaller primes for lots of divisors. See also problem 10. - Dan "the Man" Bach %;-}

Jonathan Ediyanto - 7 pts (correct number, 840; flawed solution/explanation)
Joao Sousa - 3 pts (answered 960, only 28 divisors)
Sarah DeSanctis - 1 pt - honorable mention

Problem #2 - Posted Friday, November 28, 1997
Two trains are 2 miles apart and are traveling towards each other on the same
track, each train going 30 mph. A fly going 60 mph starts at the nose of one train,
flies toward the other train, and upon reaching the second train immediately turns
around and flies back towards the first train. The fly buzzes back and forth until all
three collide. How far did the fly fly?

Solution: By Joao Sousa, DVC Calculus Student (first correct answer)
The trains will each go 1 mile in 1/30 hr = 2 min, so they collide in 2 min.
The fly flies 60 mi/hr = 1 mi / min, so in 2 min it goes 2 miles.
The fly flew 2 miles before is was frog food. - Dan %;-}

Joao Sousa - 10 pts (first correct answer, good explanation)
Peter Williams - 7 pts (Dec.1)
Sarah DeSanctis - 5 pts - (later that day)
Hoodin Hamidi - 3 pts - (just over the wire)

Problem #3 - Posted Sunday, December 7, 1997
A unit fraction is of the form 1 / n (where n = whole no. >= 1.)
#3 a) Starting with 1 / 1 , then 1 / 2 , 1 / 3 , etc., how many unit fractions
does it take to add up to more than pi ?
#3 b) Express 3 / 23 as the sum of two unit fractions; 3 / 23 = 1 / a + 1 / b .
#3 c) Write 5 / 4 as the sum of distinct unit fractions. Fine print . . .
The winner in part c) will be the one using the smallest number of fractions.
In case of a tie, the winner is the one with the "smallest biggest denominator."

Solution: (By Sarah DeSanctis)
a) 1/1 + 1/2 + . . . + 1/13 = about 3.18, so it takes 13 unit fractions to exceed pi.
b) Try 3/23 - 1/1, 3/23 - 1/2 , etc. until positive: 3/23 - 1/8 = 1/184 so 3/23 = 1/8 + 1/184.
c) Is this a trick question? 5/4 = 1/1 + 1/4. That seems too easy.
No, Sarah, it wasn't a trick question, just a mistake in my statement; I meant to say,
"How do you make 5/4 with unit fractions starting with 1/2 ?"
Then the best answer is 5/4 = 1/2 + 1/3 + 1/4 + 1/6. - - Dan - %;-}

Sarah DeSanctis - 10 pts (first correct answer, good explanation)
Peter Williams - 6 pts (1 pt penalty for too-brief expl and wrote 3/24 not 3/23))
Joao Sousa - 2 pts - (partly late, no part c)

Problem #4 - Posted Thursday, December 18, 1997
A census-taker rings Mr. Simpson's bell and asks how many children he has.
"Three daughters," he replies.
"And how old are they, in whole numbers?" asks the census-taker.
"Well, I'll tell you this: the product of their ages is 72, and the sum of their ages is my house number."
"But that isn't enough information!" complains the census taker.
"Okay, my oldest daughter (in years) likes chocolate milk," replies Mr. Simpson.
With that, the census-taker nods and writes down the three ages.
How old are the Simpson girls, and how did the census-taker figure it out?
Include a full explanation!

Solution: . . . . . . . . . . . . . . . . . . . . . . . .
Don't assume that only kids like chocolate milk, or that a "daughter" can't be 24 or 36 or even 72.
To avoid repetition we'll go youngest to oldest. (Think primes and factorizations!)
 Youngest 1 1 1 1 1 1 2 2 2 2 3 3 Next 1 2 3 4 6 8 2 3 4 6 3 4 Oldest 72 36 24 18 12 9 18 12 9 6 8 6 Sum 74 39 28 23 19 18 22 17 15 14 14 13
The key lies in the statement "But that isn't enough information!".
Since the census taker can see the house number, if it was anything but 14,
the issue would be settled. With 2, 6, 6 there would be no "oldest" daughter,
so the answer has to be : The daughters are 3, 3, and 8.

Sarah DeSanctis - 10 pts (nice job, have some chocolate milk on me!)
Joao Sousa - 2 pts - (too many assumptions made)
Peter Williams - 1 pt (see, you don't need the house number)

Problem #5 - Posted Monday, December 29, 1997
Let Sqrt(x) or -/x denote the (positive) square root of x,
as in Sqrt(100) = -/100 = 10.
Also x ^ 2 will mean x squared, as in 10 ^ 2 = 10 * 10 = 100.

1) . . If: Sqrt(m) + Sqrt(n) = 13 , and m and n differ by 65,
what is the largest possible value of m ?
Solution: m - n = (-/m + -/n)(-/m - -/n) = 65 ; (-/m + -/n) = 13 ;
(13)(-/m - -/n) = 65 ; -/m - -/n = 5 ; 2 -/m = 18 ; -/m = 9 ; m = 81.

2) . . Notice that the equation x^2 - 3 = 0 has a solution x = Sqrt(3).
Find a polynomial equation in x, with integer coefficients, having
x = Sqrt(3) + Sqrt(5) as a solution.
Solution: x - -/3 = -/5 ; (x - -/3)^2 = (-/5)^2 ; x^2 - 2-/3 x + 3 = 5 ;
x^2 - 2 = 2-/3 x ; (x^2 - 2)^2 = (2-/3 x)^2 ; x^4 - 4 x^2 + 4 = 12 x^2 ;
x^4 - 16 x^2 + 4 = 0 is our equation.
Number Theory Note: This is called the "minimal polynomial"
of the "degree 4 algebraic number" -/3 + -/5 .

3) . . What is a really good fraction approximation for Sqrt(17),
Solution: -/16 = 4 ; -/17 = 4.1231056... :=: 4.125 = 4 + 1/8 = 4 1/8 or 33 / 8.
This is good beacuse it's really close with a small denominator.
4.123 = 4123/1000 is less impressive since it's "inefficient."
(Got Calculus? Notice that on the graph y = f(x) = -/x,
the slope of the tangent line at x = 16 is m = f ' (16) = 1/(2-/16) = 1/8 = rise * 1 ,
hence the approx -/17 :=: 4 1/8.)
So, -/17 = -/(4^2 + 1) :=: 4 + 1/(2*4)
In general, -/(n^2 + 1) :=: n + 1/(2 n).
For example -/101 :=: 10 + 1/20 :=: 10.05. (Actual -/101 :=: 10.0498756...)

Number Theory Note: (33/8)^2 :=: 17 --> 33^2 :=: 17 * 8^2 --> 1089 :=: 1088
So {x = 33, y = 8} is a solution to "Pell's Equation" x^2 - 17 y^2 = 1.

Sarah DeSanctis - 5 pts (1: good job, 2:..., 3: ok but generalize)
Joao Sousa - 4 pts - (good job on #2)

Problem #6 - Posted Wednesday, January 7, 1998
The people living on Sesame Street all decide to buy new house numbers,
so they line up at the store in order of their addresses: 1, 2, 3, . . . .
If the store has 100 of each digit, what is the first address that
won't be able to buy its house numbers?

Solution: The digits are fairly balanced from house #1 to #99, but
after that the 1's are used more, so the 1,s will rum out first.
From 1-99: 1, 10, 11, . . . , 19, 21, 31, . . . , 91 (21 1's).
From 100 - 109 (11 1's). From 110 - 119 (21 1's). So far: 53.
Then 11 each for 120-129, 130-139, etc: 97 up to 159; 100 up to 162.
So #162 gets theirs but #163 is the first house not to get its number!

Sarah DeSanctis - 10 pts (credit for "it's 163")
Joao Sousa - 5 pts - (answered first, nice counting method, but off by 1. Missed it by that much!)
* The rules are 7 pts for the second correct answer, so I'm being generous. *

Problem #7 - Posted Friday, January 23, 1998

What number comes next? Give reasons!
a) 2, 3, 5, 8, 13, _?_
Answer: 21 (add two numbers to get the next - "Fibonacci Sequence")
b) 2, 3, 5, 7, 11, _?_
Answer: 13 (the next prime number; there are 25 primes under 100 and 168 under 1000.)
c) 3, 3, 5, 4, 4, 3, 5, 5, 4, _?_
Answer: 3 (the number of letters in "one, two, . . . , ten")
d) 1, 3, 7, 15, 31, _?_
Answer: 63 (each one is 1 less than the next power of 2 ; an = 2^n - 1
e) 1, 4, 27, 256, 3125, _?_
Answer: 46,656 (1^1, 2^2, 3^3, . . . , 6^6)
f) 1, 2, 6, 24, 120, 720, _?_
Answer: 5040 (the factorials; 1!, 2!, . . . , 7!, where e.g. 4! = 4*3*2*1 = 24.
n! is the product of the first n natural numbers.
Or, we could say "times 2, times 3, times 4, etc.")
g) 1, 2, 6, 30, 210, _?_
Answer: 2310 (the product of the first n-1 primes,
or, "times 2, times 3, times 5, times 7, times 11")
Answer key: a) 21 . b) 13 . c) 3 . d) 63 . e) 46656 . f) 5040 . g) 2310 .

Joao Sousa - 6 pts - (Got 4 of 7 sequences perfect - good try on 2 of the other 3.)
Leo Bach - 1 pt - (For at least asking a good question!)

Problem #8 - Posted Friday, January 30, 1998
When I have wine with a few people and we clink glasses and say "salud", I can always
tell if everyone has "clinked" with everyone else, because I know math! Let's assume
each person clinks each other person exactly once. If there are 2 people, there is one
8a. How many clinks are there for 4, 5, 6, . . . 10 people?
Answer: Each person clinks with all others, so for 4 people it's 3 + 2 + 1 = 6 clinks.
For 5 it's 4 + 3 + 2 + 1 = 10, and we can use the binomial coefficient nC2 to do each one. People:Clinks: 4:6, 5:10, 6:15, 7:21, 8:28, 9:36, 10:45.
8b. How many people were there if I heard 903 clinks?
8c. What is the formula for the quantity : c(n) = number of clinks for a group of n people ?
Answer: c(n) = nC2 = n(n-1)/2 = (n^2 - n) / 2 is "n choose 2",
which can be gotten from Pascal's Triangle or from the arithmetic series
n-1 + n-2 + . . . + 3 + 2 + 1 = n(n-1)/2 [ there are n/2 pairs that add to n-1 ]. Now to finish #8b...
nC2 = n(n-1)/2 = 903, (n^2 - n) = 1806, n^2 - n - 1806 = 0, (n - 43)(n + 42) = 0 ,
n = 43 or -42, so there were 43 people at the clinkfest.

Joao Sousa - 5 pts - (Be more thorough - and 42 was close but no cigar with your wine.)

Problem #9 - Posted Sunday, February 15, 1998
9a) If 3 ^ a = 4 and 4 ^ b = 8, what is 9 ^ (a - b) ?
Answer: 9^a = (3^2)^a = 3^(2a) = (3^a)^2 = (4)^2 = 16;
Let's find b: 4^b = (2^2)^b = 2^(2b) = 8 = 2^3 , so 2b = 3 ; b = 3/2.
So 9 ^ (a - b) = (9^a) / (9^b) = 16 / (9^b) = 16 / (9^(3/2)) = 16 / (3^3) = 16/27.
9b) Can you find numbers a =/= b such that a ^ b = b ^ a ?
Answer: Sure, 4^2 = 2^4 = 16, but did you know there is a whole curve of (a,b) in the ab-plane with a^b = b^a ?! Use an implicit plot to graph x^y - y^x = 0 ; it goes through any (a,a); also (2,4), (4,2), and a host of other points such as (2.478,3) approx.
Check this : 2.478^3 :=: 3^2.478. (The :=: means "approx equal")
An answer of "no" might be considered correct, if you couldn't find the numbers!
9c) If a \$ b means a ^ b - b ^ a , what is 4 \$ 6 ?
Answer: 4^6 - 6^4 = 4096 - 1296 = 2800. Note a \$ b is often positive if a < b. Is it always?
Is there a simple condition for when a \$ b > 0 ?
9d) Is 2 \$ (3 \$ 4) the same as (2 \$ 3) \$ 4 ?
Answer: No. Here's a proof that \$ is "not associative" :
2 \$ (3 \$ 4) = 2 \$ (3^4 - 4^3) = 2 \$ (81 - 64) = 2 \$ 17 = . . .
. . . = 2^17 - 17^2 = 16384 - 289 = 130783 , while
(2 \$ 3) \$ 4 = (2^3 - 3^2) \$ 4 = (8 - 9) \$ 4 = (-1) \$ 4 = . . .
. . . = (-1)^4 - 4^(-1) = 1 - 1/4 = 3/4 . That's very different!

Andy Murdock - 8 pts - First answer was 16/27 , not your approx value of 0.593.
Joao Sousa - 4 pts - Your 9a) worked out to be 0.4103, and 9c) wasn't finished.

Problem #10 - Posted Saturday, February 28, 1998
A natural number is abundant if its proper divisors (not including itself) add up to more
than the number. (For example: 12 is abundant because the divisors of 12 are 1, 2, 3, 4, 6, and 12,
and 1 + 2 + 3 + 4 + 6 = 16 > 12.) Notice 6 = 1 + 2 + 3 ; 6 is called perfect.
Also 15 is deficient (not abundant or perfect) because 1 + 3 + 5 = 9 < 15.
Hint: Numbers like 12 = 2 * 2 * 3 . that have lots of small prime factors, tend to be abundant.

Solution: The hint means that powers of smaller primes give smaller numbers with lots of divisors that might add up to more than the original number.
Divisors of powers of 3 never add up: 1 + 3 + 9 = 14 < 27, etc. So 3^b isn't enough.
(Note 14/27 :=: 1/2, the limit. We want this ratio to exceed 1. Good job, Andy.)
Try 3^b * 5^c: n = 3*5 = 15: 1 + 3 + 5 = 9 < 15; 9/15 = 0.6
n = 45: 1 + 3 + 5 + 9 + 15 = 33 < 45; 33/45 :=: 0.733
(For n = 3^6 * 5^4 = 455,625: sum of proper div = 398,008 ; ratio :=: 0.874)
This seems to fall short of our goal of 1, so we bring in powers of 7.
n = 3^2 * 5 * 7 = 315; sum of all div = (1+3+9)(1+5)(1+7) = 624,
sum of proper div = 624 - 315 = 309, just missed!
n = 3^3 * 5 * 7 = 945; sum of all div = (1+3+9+27)(1+5)(1+7) = 1920,
sum of proper div = 1920 - 945 = 975 , it's abundant!
The answer is: The smallest odd abundant number is 945.