Trotter Dates

Naming numbers according to some property or characteristic that they possess is a legitimate — albeit egotistic — activity of some mathematicians. (See Keith Numbers for a simple and elegant example.)

Another interesting name category that I have recently learned about is called “Niven Numbers”. These are merely numbers that are divisible by the sum of the digits of the number. [Or to put it another way: the sum of the digits is a factor of the number itself.] A simple example should suffice for our purposes here. 126 is a Niven number because 1 + 2 + 6 = 9, and 126 divided by 9 is 14.

So I decided to combine the idea of “Product Dates” (the date part) with that of Niven numbers (the divisibility part) and came up with a new category of numbers, or dates, for all my Trotter Math friends of the world: TROTTER DATES.

A “Trotter Date” shall be defined to be a date for which the year number — either the short 2-digit form or the full 4-digit form — is divisible by the sum of the month number and the day number.

Let’s take an example, my own birthday: May 21. In 1978 we would have written this as 5/21/78. And 78 is divisible by 26, the sum of 5 and 21. Hence, that date shall be considered as a “Trotter Date”.

Of course, my birthday produces two other TD’s (Trotter Dates) in a given century, namely 5/21/26 (I wasn’t alive for that one!) and 5/21/52 (I was alive for that one, but was not aware of its signifigence at the time.)

Now if we go for the full 4-digit form using my birthday, there are some other TD’s awaiting us. For example, 5/21/1976 was my latest personal TD because 1976 divided by 26 is 76! (Hmm…, now that’s a nice coincidence.)

My next one will occur on 5/21/2002 because,… well, you understand this by now don’t you? (By the way, that date will be significant for me and my family; 2002 is the year my son graduates from high school!)
Searching for more TD’s now can be a nice activity in the elementary or middle school classroom.

~More about this topic on a later day…

May 28, 1998…

Hey, today in one of my Pre-algebra classes one of my students, Estefania Lopez, showed me a Trotter Date: her birthday, in fact, the day she was born! She used the “long” form of the date: March 13, 1984. So she wrote: 3/13/1984. Can you see why it is a Trotter Date Birthday (TDB)? Congratulations, Stefy!

Then later today another student, Andrew Kranstover, discovered that his date of birth was a TD as well. He was born on November 20, 1984. He also used the long form: 11/20/1984. Gee, isn’t it exciting to find two TDB’s in one day!!!

TEACHER’S GUIDE:

One suggestion to make this activity into a true “pre-algebra” item would be to present the process in the following manner:

Tell the students that here is a “formula” in which they will substitute certain values:

(m + d)x = y

where m = month number, d = day number, and y = year number. Then x, the solution of our equation, will be obtained by regular “algebraic” procedures. If it comes out as an integer, then we have a TD; otherwise not.

For example, let’s look at April 5, 1998. In the long form, we have 4/5/1998. So m = 4, d = 5, and y = 1998. By substituting, we have

(4 + 5)x = 1998
9x = 1998
x = 222

Wow! x is an integer! So, we have a TROTTER DATE.

The short form of the year (4/5/98) does not produce a TD. Observe:

(4 + 5)x = 98
9x = 98
x = 10.888…

In this way, we have raised the value of this activity to one that is getting ready for algebra, personalized (one’s birthday can be used), and deals with simple ideas in the regular math curriculum.

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