Niven Numbers

The topic of Niven Numbers is briefly mentioned on another page in this website. But since writing that, I have been working on the concept a little bit more with some students and I now feel that it deserves its own separate page. So what follows here is a discussion about how it can be utilized in the regular school math class.


We begin by defining once again the concept of what a Niven Number is:

A Niven Number is any whole number that is divisible by the sum of its digits.


The only real trouble with this is the meaning of the word divisible. In number theory it simply means “division with no remainder“. Another way to express that is to say that the divisor in the division process is also a factor of the dividend.

An example or two should make all that quite clear. Let’s take the number 126. First, we must find the sum of the digits.

1 + 2 + 6 = 9

Next, we try to divide 126 by 9. And we find…

126 ÷ 9 = 14

There it is. 14 and no remainder; or put another way: 9 is a factor of 126 because

126 = 9 × 14

So 126 is a NIVEN NUMBER.

On the other side of the coin 121 is not a Niven Number. Its digit sum of 4 is not a divisor (or factor) of the number itself. 121 divided by 4 yields a quotient of 30 and a remainder of 1.

Doesn’t sound too hard now, does it? Well, it all depends. For young students this has proved to be sufficiently challenging, mostly because it is all so different from the usual math fare that is served up to them by their textbooks and teachers.

For example, I have posed the following simple question to students:

How many Niven Numbers can you find from 100 to 200?

It has been interesting to observe their initial reactions. Mainly they don’t quite know where to begin, what numbers to pick, and so on. They seem to be overly concerned about being unable to state the number of Niven Numbers in that range in a short amount of time, rather than realizing that it will take some time and patience to arrive at it. Next, they start out in an unorganized way, choosing numbers at random to test, rather than “beginning small and working up”.

So, after allowing them to struggle with the whole matter for a while, I make the suggestion, “Why not make a list of all the numbers from 100 to 200, inclusive; then start at the 100 and work your way up? We can circle the Niven Numbers, and cross off those that are not.” This shows a good problem solving strategy that many had never thought of. And at the same time, it allows us to review the divisibility tests that the students had studied earlier, but for lack of use in any meaningful context, had forgotten.

This organized strategy also has a unique advantage. One begins the find some easy Niven Numbers rather early: 100, 102,
108, 110, 111, and 112. Nothing breeds confidence and understanding better than some early success.

Furthermore, in this range of 13 numbers (100-112) there are several good opportunities for discussing another number property that aids our search. You see, there is no real need to carry out the long division and find the quotient IF the main objective is to determine if a number is Niven or not. If the number in question is odd and the digit sum is even, it is clear that the number can be crossed off the list. For example, 101. It is odd, but its digit sum is even. No odd number can be divided by an even number without leaving a remainder of some sort. Or to put it another
way:

No odd number is the product of an even number and something else.

So numbers like 101, 103, 105 and 107 can be eliminated rather easily. Another simple case arises when numbers have a digit sum of 10 and the number itself does not end with zero, as is the case for 109. Finally, in this limited range that we are considering, 104 is of interest. It has a digit sum of 5, yet the number does not end with a zero or 5. Basic and obvious as these points may seem to some, they still need to be brought into the discussion.

So with all these powerful ideas and strategies working for us now, just how many number from 100 to 200 are indeed Niven? We intend to leave that up to you to find. It’s not nice to spoil your “thrill of victory and agony of defeat”, right?


Where next?, you might be asking yourself now. Well, several ideas come to my mind; perhaps you could dream up some of your own. How about these two?

  1. What is the largest 3-digit Niven Number? (Also, the 2nd largest and 3rd largest ones?)
  2. What is the smallest Niven Number formed by the digits “1”, “6”, and as many zeros as needed?

Of course, we should mention the natural extension of the problem presented above: How many Niven Numbers are there in each “CENTURY” from 0 to 999? That is to say: 0-99, 100-199, 200-299, …, 900-999. The point here is to investigate these groups of one hundred numbers to see if there are any interesting patterns. Also to determine which century has the most (or the fewest) Niven Numbers. [Perhaps a bar graph could be the final presentation format?]

This would not be as tough a job as it might first appear if calculators, or even computers, were utilized. After all, it is the thinking, the mathematical investigation that is important here, not one’s ability at looonngg division. And don’t worry if your calculator doesn’t give division answers in “quotient-remainder” style. [Nowadays some do have that feature.] Any ordinary model will suffice, because of this basic fact:

If the division would produce a remainder in the traditional sense, then the calculator will give a “decimal” result instead.

Using our example earlier of 121, we would have this:

121 ÷ 4 = 30.25

And that is sufficient to tell us that there had to be a remainder of some sort, so 121 is therefore not Niven. (For the teacher and student willing to delve deeper into the matter, a discussion could be held about the relationships between ordinary remainders and the decimal part of the result. Thus making this project into a worthwhile learning experience.)

It’s now time to “get Niven”, and find those NN’s out there.

Good luck, and let me me know what you find.


POSTSCRIPT 1:

After writing the above article, I received an e-mail from the individual who first introduced me to this topic, a math professor in Colorado named Monte Zerger. Here I present selected portions of what he told me:

“You got me thinking about Niven numbers again, so I
did a little quick research. According to an article in the
Journal of Recreational Mathematics the origin of the name
is as follows. In 1977, Ivan Niven, a famous number theorist
presented a talk at a conference in which he mentioned
integers which are twice the sum of their digits. Then in
an article by Kennedy appearing in 1982, and in honor of
Niven, he christened numbers which are divisible by their
digital sum “Niven numbers.”

In most all the literature I have seen this is the accepted
name. However, I note that in his “Dictionary of Curious and
Interesting Numbers” Wells calls them “Harshad numbers.” …

In an article appearing in Journal of Recreational Mathematics in 1997, Sandro Boscaro defines and discusses Niven-morphic numbers. These are Niven numbers which terminate in their digital sum. For example 912 is such a number since it has a digital sum of 12, is divisible by 12, and ends in 12. In this article he proves a startling result. There is no Niven-morphic number which has a digital sum of 11, is divisible by 11 and ends in 11. 11 is the *only* such exception.

Of course 1, 2, 3, …,9 are all trivially Nivenmorphic. I played around this morning with a simple BASIC program and found the smallest (I think!) Nivenmorphics for 10, 12, 13, …, 19. They are [answers withheld to give you the chance to find them. Sorry, that’s the way things are in the World of Trotter Math. :=) ]”

POSTSCRIPT 2:

Later our correspondence grew a bit more, and the following curious trivia resulted:

  1. In my investigation for NNs under 1000 in order to understand what would be involved for the “century” question above, I found a string of 4 consecutive ones: 510, 511, 512, 513. And that was the longest string in that range.
  2. To which Monte replied, “My computer program found three other strings of four terms: 1014-1017, 2022-2025, and 3030-3033. I still like the string starting at 510 the best since 510 is the Dewey Decimal classification for mathematics and “cloning” it we have the product of the first 7 prime numbers as well as the product of the 7th through 10th Fibonacci numbers.”
  3. And Monte’s BASIC program uncovered the following item: “The smallest string of five consecutive Nivens begins with 131,052. I checked all the way up to 10,000,000 and could not find a string of six.”

    (Well, I think that’s going high enough for most purposes with young students in school. Perhaps a worthwhile activity would be to ask them to confirm that the numbers mentioned above are indeed Niven Numbers.)

    Then later Monte came through with a “really big” piece of information: “Ran across something today that will interest you. It has been proven there cannot be a string of more than 20 consecutive Nivens. A string of 20 has been found. Each member has 44,363,342,786 digits.”

  4. I also thought of the reverse side of things… How about the idea of strings of non-NNs? Between what two NNs can we find great numbers of non-NNs? In my limited, yet accessible-to-young-students list, I found these 17-term “droughts”: 558-576, 666-684, 736-756, 846-864, & 972-990. No Nivens here!
  5. Here’s more from the mind of Monte: “I suppose you noticed that whereas the current decade had only one Niven year (1998), the next decade has four! 2000, 2001, 2004 and 2007.”

[Stay tuned, dear reader. There’s no telling where this is going to end!]


Well, it took awhile, but there is some new action about Niven Numbers. On May 17, 2001, I received the following email:

I just noted your page on “Niven Numbers” and enjoyed it…………………I’m still very much interested in such numbers. Ivan Niven died last year but at least his numbers will help people to remember him. By the way, have you investigated Niven numbers in differenct bases? Have you considered which powers of two are Niven?

Sincerely,

Robert E. Kennedy

Ok! Different bases and powers of two… hmm… Looks like we have
more work to do.


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