# A “Mean” Product

My mother once taught in a small country three-room schoolhouse, ampoule where students of different age levels often had to study the same course together. For the course in American History, salve the kids in the 6th, 7th, and 8th grades were combined into one class. This means that their ages were between 10 and 14.

If the product of their ages was 41,760,576, what was the average, or arithmetic mean, of their ages?

# Same Digits Multiplication

Here is a famous old multiplication idea. Look at the two simple problems below.

```	       	      51		    21
× 3		   × 6
153		   126
```

The strange property here is that the digits that were used in the factors are the only ones that appear in the product! And even stranger is the fact that try as hard as you can, you cannot find anymore cases like this—for THREE digits, that is. But when you go to four or five digits, well, that’s a different story, as you will soon see.

In the problems below some of the digits are missing, and *** are put in their places. But they all obey this rule: the digits used in the factors are the ONLY ones that show up in the product, and vice versa. So for this lesson you are supposed to find the missing digits.

```        A)    473	      B)    ***		   C)    4128
×  8	   	   ×  9		        ×   3
****		   3159		        *****

D)    ****	      E)   ****		   F)    ****
×   3	          ×   6		        ×   8
12843	          15246		        39784

G)   7***	      H)   **2*		   I)    **51
×   9	          ×   3		        ×   *
6*149	          *1375		        2*7*3

J)   **14	      K)   ****		   L)    ****
×   *	          ×   *		        ×   *
*7*28	          17482		        51268```

Now we will do the same idea, but here we will just use two-place numbers. Below are the only eight numbers that you are supposed to use to make four problems. How they should be paired off and still obey our rule of same digits in the product? And what are the four correct products?

`         15     21     81     93     35     87     27     41`

Source for idea: H. E. Dudeney, Amusements in Mathematics. Dover, 1958. pp. 15 & 156.

tt(1/23/77)

# Consecutive Digit Products (CDP’s)

Part I

Carefully find the products of these multiplication exercises.

(A)   41  × 3

(B)   576 × 6

(C)   11728 × 2

If you did your work correctly, you should have noticed that the digits in each product were “in order”. That is rather interesting, don’t you think?

When the digits appear “in order” like that, we say they are consecutive digits.

Here are some more problems. Each one has a “consecutive digit product (CDP)”.

(1)     81 × 7            (6)     18 × 13
(2)     263 × 3          (7)     335 × 7
(3)     57 × 8            (8)     64 × 54
(4)     23 × 15          (9)     219 × 31
(5)     27 × 21         (10)     167 × 34

tt(8/4/82)

Teacher’s Note: This and the following three activity pages were written before calculators were commonly accepted for use in the classroom. Hence, you should take this into consideration when using this material.

The fundamental purpose behind this seeming “drill & kill” sort of activity is actually the opposite. Namely, once you have multiplied the numbers, you should have, if done correctly, a surprise awaiting you: a product with some interesting aspect, in this case consecutive digits of one form or another. In this way, a reward of sorts is provided while at the same time practicing the old fashioned (or, if you prefer, time honored) skill of multiplying whole numbers. And not to be overlooked is the encouragement of the habit of examining one’s answer to see if it is reasonable; in this activity the focussing on the answer is merely to see if it meets the criteria of the lesson. But in general the habit of “looking back” is one that is not well established in the minds of many and needs to be promoted more.

tt(6/5/98)

Part II

Here are some more problems that have “consecutive digit products”. But this time there is a difference. What is it?

(A)   48  × 9

(B)   727 × 9

(C)  18107  × 3

Of course, you see that the digits are still in order, but this time their order is reversed!

Try the following exercises to see yet another type of answer.

(D)   27 × 5

(E)  617 × 4

(F)   3251 × 3

In those problems the digits are either odd or even, while at the same time they are still consecutive.

Now, all the problems below use these ideas. But, which is which?

(1)     107 × 3             (6)     443 × 17
(2)     181 × 3             (7)     149 × 58
(3)     51 × 15             (8)     89 × 86
(4)     679 × 8             (9)     149 × 29
(5)     1193 × 3          (10)     59 × 23

tt(8/4/82)

Part III

Here are twenty more problems. Most of them have products of consecutive digits; but some of them do not! Can you find those that do not have CDP’s?

(1)     617 × 2           (11)     73 × 12
(2)     47 × 21           (12)     197 × 45
(3)     36 × 12           (13)     823 × 12
(4)     128 × 27         (14)     2932 × 8
(5)     65 × 19           (15)     2659 × 13
(6)     93 × 73           (16)     953 × 57
(7)     76 × 31           (17)     178 × 43
(8)     97 × 56           (18)     733 × 32
(9)     298 × 29         (19)     557 × 8
(10)     24 × 19         (20)     138 × 331

tt(8/4/82)

Part IV

This lesson presents several new ideas. Two of them are: number palindromes and number tautonyms.

Perform the multiplications to see these interesting types of products–and others–come out.

In all of them, however, the idea of consecutive digits is still there.

(1)     3261 × 14             (11)     945 × 143
(2)     333 × 37               (12)     3717 × 143
(3)     353 × 91               (13)     2354 × 273
(4)     8182 × 8               (14)     902 × 273
(5)     407 × 333             (15)     3435 × 33
(6)     6105 × 87             (16)     2277 × 243
(7)     3737 × 66             (17)     537 × 418
(8)     9731 × 66             (18)     20134 × 33
(9)     443 × 223             (19)     20219 × 6
(10)     2442 × 263         (20)     50471 × 3

tt(8/7/82)

# 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.

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.

# Product Dates

[The following item appeared in the “From the File” section of The ARITHMETIC TEACHER, October 1983, p. 53. Later it was referenced in other NCTM publications.]

Here is a novel activity that can be used with selected dates—for example, March 27, 1981—which might be introduced to the class in the following way:

“Today is a special day for mathematicians. To see why, let’s write the date in the usual brief form, 3/27/81. Notice that we can also state this as a multiplication sentence, 3 × 27 = 81. Interesting, wouldn’t you say! Let’s call this a product date. Are there any more product dates in 1981?”

After students have had time to explore this question and find answers, the following questions can be posed:

1. Did last year have any product dates? If so, when?

2. Will next year have any product dates? If so, when?

3. Did your birthday fall on a product date?

4. How many product dates are there in any specific decade, say, the 1970s, the 1960s, and so on. List them.

5. The BIG question: How many product dates are there in a whole century? List them.

From the file of Terrel Trotter, Jr., McKinley Middle School, Harvey, IL 60426 (who found the idea in the Journal of Recreational Mathematics, April 1969 and October 1972)
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