Posts tagged ‘factor’
7 Math Mistakes to be Aware Of
April is Math Awareness Month, and some things to be aware of this month — as well as the whole year through — are common math errors. Here are seven that show up frequently.
Incorrect Addition of Fractions. It’s common for kids to add fractions as follows:
And while that algorithm works for batting averages in baseball, it doesn’t work in most other places. More importantly, this mistake is often unaccompanied by reasoning. For example, a student who claims that 2/3 + 4/5 = 7/9 doesn’t realize that with each addend greater than 1/2, then the sum should be greater than 1. That lack of thought bothers me.
Cancellation of Digits, Not Factors. While it’s true that 16/64 = 1/4 and 19/95 = 1/5, students who think the algorithm involves cancelling digits may also argue that 13/39 = 1/9, and that just ain’t right.
Incorrect Distribution. This one takes a lot of forms. In middle school, kids will say that 4(2 + 3) = 8 + 3. As they get older, they apply the distributive property to exponents and claim that (3 + 4)^{2} = 3^{2} + 4^{2} or, more generally, that (a + b)^{2} = a^{2} + b^{2}.
The Retail Law of Close Numbers. A large portion of the population will buy a shirt for $19.99 that they’d pass up if it had a price tag of $20.00. Even though the amounts only differ by one cent, a lesser digit in the tens place makes the price feel much lower. Crazy, but true.
Ignoring the Big Picture. If you are a driver who is interested primarily in speed (and less concerned with price, looks, fuel efficiency, or other factors), would you rather have a vehicle with 305 horsepower or one with 470 horsepower? If you chose the latter option, congratulations! While the owner of a sweet 305hp Ford Mustang will be sitting at home and sipping a mint julep on his front porch, you’ll still be doing 30 mph on the highway in your Sherman tank.
Correlation Implies Causation. As ice cream sales increase, the number of drowning deaths increases, too. But that doesn’t mean that having an ice cream cone willl make you less likely to swim safely, even if you failed to heed your mother’s advice to wait 30 minutes after eating. It’s just that ice cream sales and swimmingrelated deaths increase in summer, both of which are to be expected.
Just because two things happen to coincide doesn’t mean that one is the direct (or even indirect) result of the other.
Percents Don’t Work That Way. A 20% decrease followed by a 20% increase does not return you to the initial value. If you invest $100 in a company, and it loses 20% the first year, your investment will then be worth $80. If it gains 20% the next year, you’ll now have $96. Uhoh.
What common math error do you see frequently, and which one bothers you the most?
Two Simple Math Games
In his article “What Is the Name of This Game?” author John Mahoney discussed the mathematics of the following game:
Cards numbered 19 are placed face up on a table. Two players alternate picking up one card at a time. The winner is the first player who has exactly three cards with a sum of 15.
You can play this game with nine cards removed from a deck of cards, or you can play online by going to http://illuminations.nctm.org/deepseaduel. The online version is a oneplayer game, but it has modifications that use different numbers of cards, different values on the cards, and different required sums.
Can you find the winning strategy for this game? (Hint: The strategy is described in the linked article above.)
Here’s a modification of the game that seems interesting, too.
Use cards with the following numbers: 1, 2, 3, 4, 6, 9, 12, 18, 36. The winner is the first player who has exactly three cards with a product of 216.
The optimal strategy for this game is different than the strategy from the original game. Can you find it?
Note: For the original game, there are eight sets of three cards with a sum of 15:
{1, 5, 9}, {1, 6, 8}, {2, 4, 9}, {2, 5, 8},
{2, 6, 7}, {3, 4, 8}, {3, 5, 7}, {4, 5, 6}
For the modification, there are 36 sets of three cards with a product of 216.
Perhaps that fact will help you identify the optimal strategy.
What is Your Favorite Number?
The WordPress PostAWeek Challenge sends me a daily topic idea to consider for blog posts. Often, the prompts are not appropriate for a math jokes blog. For instance, some recent prompts have been:
 Grab the nearest book (or website) to you right now. Jump to paragraph 3, second sentence. Write it in a post.
 How do you find your muse?
 If you could bring one fictional character to life for a day, who would you choose?
But today’s prompt landed in my wheelhouse:
What is your favorite number, and why?
When Art Benjamin appeared on the Colbert Report, he said that 2,520 was his favorite number when he was a kid. When Stephen Colbert asked him why, he replied, “It was the smallest number that was divisible by all the numbers from 1 through 10.”
Tonight, I asked my twin sons Alex and Eli what their favorite numbers are.
Eli: 5, 15, 55, because my favorite number is really 5, but 15 and 55 are triangular numbers that have 5’s in them.
Alex: 21, because my favorite numbers used to be 1 and 2, and because it’s the number of cards you deal when we play Uno (3 players, 7 cards each).
My favorite number is 153, for lots of reasons:
 It is the smallest nontrivial Armstrong (or narcissistic) number — that is, it is an n‑digit number that is equal to the sum of the nth powers of its digits: 1^{3} + 5^{3} + 3^{3} = 153.
 Its prime factors are 3 and 17, and my birthday is 3/17.
 It is a triangular number. (Consequently, it’s the sum of 1 + 2 + 3 + … + 17.) As 351 is also a triangular number, 153 is also a reversible triangular number.
 It is the sum of the first five factorials: 1! + 2! + 3! + 4! + 5! = 153.
 The sum of its digits is 9, and the sum of its proper divisors is 9^{2}.
 It is one of only six known truncated triangular numbers, which means that 1, 15 and 153 are all triangular numbers.
Mathematician John Baez claims that his favorite numbers are 5, 8, and 24.
Got a favorite number? Share it, as well as the reason it’s your favorite, in the comments.
Fun Factor
Two numbers were having a conversation about their social lives.
28: Did you hear that 284 broke up with 220?
6: I’m not surprised. He’s far from perfect. But at least their breakup was amicable.
28: Yeah, well, I heard she started seeing 12.
6: Really? He doesn’t have abundant charm. Don’t you think 10 would be a better match for her?
28: I don’t know. He seems so solitary!
Speaking of factors, I learned a neat trick this weekend for finding the sum of the factors of a number. Before I share that, consider the method for determining how many factors a number has. Take the number 12, for instance. The prime factorization of 12 is:
12 = 2^{2} × 3
The following array can be used to generate all of the factors of 12:

2^{0}  2^{1}  2^{2} 
3^{0}  1  2  4 
3^{1}  3  6  12 
It’s obvious from the array that there are six factors. But the trick is to notice that each factor in the array is made from a power of 2 times a power of 3 — that is, each factor is equal to 2^{m} × 3^{n}, where 0 ≤ m ≤ 2 and 0 ≤ n ≤ 1. Since there are three possible values of m and two possible values for n, then there are 3 × 2 = 6 factors of 12.
In general, if the prime factorization of the number takes the form a^{p} × b^{q} × c^{r}, then the number of factors is (p + 1)(q + 1)(r + 1) for exponents p, q, and r. (The process could obviously be extended if there are more than three prime factors.)
But look at the array again. The sum of all factors of 12 is equal to sum of all products that occur within the array. However, there is an easy way to find that sum, by taking advantage of the distributive property. The sum of the powers of 2 along the top is 2^{0} + 2^{1} + 2^{2} = 7, and the sum of the powers of 3 along the left side is 3^{0} + 3^{1} = 4. Consequently, the sum of all factors of 12 is equal to:
(2^{0} + 2^{1} + 2^{2})(3^{0} + 3^{1}) = 7 × 4 = 28
In general, if the prime factorization of a number is a^{p} × b^{q} × c^{r}, then the sum of the factors is:
(a^{0} + a^{1} + … + a^{p})(b^{0} + b^{1} + … + b^{q})(c^{0} + c^{1} + … + c^{r})
And again, this could be extended if the number had more than three prime factors.
Cool, huh?
Confounding Factors
My colleague Julia is preparing a talk about factoring for an elementary audience, and she created the following problem to use as a warmup:
Take a two‑digit number ab, and find the least common multiple of a, b, and ab. For example, if you take the number 35, then LCM(3, 5, 35) = 105. For which two‑digit number ab is LCM(a, b, ab) the greatest? (The notation ab is used to indicate the two‑digit number with tens digit a and units digit b, which is equal to 10a + b. This notation is used to distinguish the two‑digit number ab from the product ab.)
Here are some math jokes about factors:
What do you call an amount that exactly divides a recipe for a sweet confection?
A fudge factor.What do algebra equations and British television have in common?
An X Factor.
Sadly, both of those are my original jokes. Sorry. To cleanse your palate, check out one of Randall Munroe’s original jokes about factoring: