Tests for Divisibility in All Number BasesAuthor(s): Richard EnglishSource: Mathematics in School, Vol. 14, No. 2 (Mar., 1985), pp. 6-7Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213964 .

Accessed: 22/04/2014 08:22

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 5.198.113.170 on Tue, 22 Apr 2014 08:22:31 AMAll use subject to JSTOR Terms and Conditions

f f

blit

in

r

r by Richard English, Wolfreton School, Hull

Some months ago while preparing a Base 5 arithmetic question sheet I was faced with the problem of trying to choose numbers which were exactly divisible by n so that there would be no remainder when the division sum was performed. When working in Base 10 this problem is easily overcome by applying the commonly known(?) tests for divisibility, but in other number bases trial, error and adjustment are necessary unless tests for divisibility are available. This prompted me to devise such tests for use not only in base 5 but in all number bases up to 10. This task proved to be a very interesting one and one which I felt could be shared with pupils so as to stimulate the develop- ment and application of ideas in the classroom.

Presenting the Task The following lesson was conducted with an able fourth year group and turned out to be a very worthwhile and interesting experience. The lesson and its aims can be divided into three parts.

Establish the common tests for divisibility in base 10. Explain why these tests work. Apply the same principles to produce tests for divisibility in all number bases up to base 10.

Common Tests for Divisibility When I asked the class to tell me all the tests for divisibility that they knew I was very surprised at the poor response. I had to rely on 2 to 3 pupils to tell me the tests while the rest of the class listened and learned. When all of the tests from 2 to 10 (excluding 7 which was omitted) had been estab- lished, a show of hands revealed that most of the class did not know more than a couple of these and so had learned something already.

Explanation I then asked the pupils to offer explanations of these tests.

Again I was surprised by the initial poor response but once the underlying principles had been recognised, the explana- tions were more forthcoming. Tests for divisibility by 10, 5, 2, 4 and 8 were explained successfully in terms of factors of the base 10 column headings. Divisibility by 9 and 3 were more difficult to explain and so I eventually had to step in and offer help.

I explained the test for divisibility by 9 by working out the remainders when each of the base 10 column headings are divided by 9. The remainder is of course always 1 (e.g. 10-9=lrl) and so any digit in base 10 tells us the remainder when the amount that the digit represents is divided by 9. For example, a 6 in the "thousands" column means that 6000 - 9 gives a remainder of 6. If the sum of all the remainders (i.e. the digits) is divisible by 9 then the number must be divisible by 9.

Once this had been explained the class was able to give me a similar explanation of the test for divisibility by 3.

Tests in other Number Bases For the third part of the lesson the class were asked to establish tests for divisibility in other number bases, work- ing in small groups, each group being assigned a number base to work in. I asked for tests by all numbers up to and including the value of the base being used (e.g. in base 5 devise tests for divisibility by 2, 3, 4 and 5).

Successful Strategies Two different approaches seemed to be adopted:

List several base 10 numbers known to be divisible by n, change them to the required base and then inspect for any common characteristics. Produce a theorem after examining the column headings of the required base and then put the theorem to the test by experimenting with a series of numbers.

Strategy One The first approach sometimes produced correct statements about the characteristics of a particular set of multiples but these were not really tests for divisibility because the characteristics mentioned were not always unique to the set of multiples concerned. For example one pupil concluded that in base 8, all multiples of 6 end in 0, 2, 4 or 6. This is correct but not all base 8 numbers that end in 0, 2, 4 or 6 are divisible by 6.

Another common error was simply to not have examined enough multiples and thus produce a test which only worked for those numbers that had been listed.

In cases like these I pointed out a number which had the required characteristics but was not a member of the set of required multiples. This led to the re-thinking and modifi- cation of the pupils' theorems until they were rigorous tests for divisibility.

Strategy Two The second approach was used successfully by a large proportion of the class to obtain a lot of the tests for divisibility and I was very impressed with the way they applied the principles we had discussed earlier in the lesson.

During the last ten minutes of the lesson we collected together all the results, tabulated them on the blackboard and discussed them briefly. A similar summary is shown in Table 1.

The pupils devised all of the tests themselves with the exception of those marked * which I have added myself. These starred tests are not quite so straightforward to devise but all are based on the remainders which are left when the column headings of the base are divided by the number concerned. The test for divisibility by 5 when working in base 8 is now given as an example.

6 Mathematics in School, March 1985

This content downloaded from 5.198.113.170 on Tue, 22 Apr 2014 08:22:31 AMAll use subject to JSTOR Terms and Conditions

Table 1 Summary of Tests for Divisibility

BASE 10 BASE 9 BASE 8 BASE 7 BASE 6 BASE 5 BASE 4 BASE 3 BASE 2

Divisibility last digit digit sum. last digit digit sum last digit digit sum last digit digit sum last digit 0 by 2 even even even even even even even even

Divisibility digit sum last digit start at digit sum last digit start at digit sum last digit 0 by 3 divisible 0, 3 or 6 units end, divisible 0 or 3 units end, divisible

by 3 add and by 3 add and by 3 subtract subtract alternate alternate digits. Sum digits. Sum divisible divisible by 3 by 3

Divisibility number digit sum last digit start at * Double * digit sum last digit 0 by 4 formed divisible 0 or 4 units end, 2nd last divisible

by last by 4 add and digit, add to by 4 2 digits subtract last. divisible alternate Sum by 4 digits. Sum divisible

divisible by 4 by 4

Divisibility last digit start at start at * start at * digit sum last digit 0 by 5 0 or 5 units end, units end, units end, divisible

add and multiply multiply by 5 subtract digits by digits by alternate 1, - 2, 1, 2, digits. -1,2. -1, -2. Sum Sum Sum divisible divisible divisible by 5 by 5 by 5

Divisibility test for test for test for digit sum last digit 0 by 6 2 and 3 2 and 3 2 and 3 divisible

by 6

Divisibility See Text See Text digit sum last digit 0 by 7 divisible

by 7

Divisibility number digit sum last digit 0 by 8 formed divisible

by last by 8 3 digits divisible by 8

Divisibility digit sum last digit 0 by 9 divisible

by 9

Divisibility last digit 0 by 10

The base 8 column headings are:-

262144's 32768's 4096's 512's 64's 8's Units

The remainders when each column heading is divided by 5 are:-

4 3 1 2 4 3 1

These remainders need to be multiplied by the digit value in that particular column, added together and then tested for divisibility by 5. So for example, the base 8 number 41737, working from the right, gives a remainder of

(7 x 1) + (3 x 3) + (7 x 4) + (1 x 2)+(4 x 1)= 50

5010 is divisible by 5 so 41737, is also divisible by 5.

Using the multiplier sequence 1, 3, 4, 2 can generate quite large numbers but we must avoid making the test for divisibility more tedious than the actual division sum we are replacing. We can however, simplify the test by a slight modification.

The 8's column gives a remainder of 3, but if we wanted to get another "bundle of 5" out of it we have a deficit of 2,

or, put another way, the remainder is - 2. Similarly the 64's column gives a remainder of 4, which is in effect a remain- der of -1 if an extra "bundle of 5" were taken. Our modified multiplier sequence, starting at the right, is there- fore 1, -2,- 1, 2. Using the previously quoted base 8 number 41737, this gives a remainder of:

(7 x 1)+(3 x - 2)+(7 x - 1)+(1 x 2)+(4 x 1)

=7-6 -7 + 2 + 4

=0

Since 0 is divisible by 5 then 41737, is also divisible by 5. There are two gaps in the summary table, both concern-

ing tests for divisibility by 7. Several tests for divisibility by 7 in base 10 have been documented in the past but I do not consider any of these to be particularly useful since the tests are more time-consuming than the division they are sup- posed to replace. Similarly when working base 9 I have been unable to find a relatively simple test. The best that I have come up with is to use the digit multiplier sequence 1, 2, - 3, starting at the right, and to test the sum of these for divisibility by 7.

Conclusions This was a very interesting and worthwhile exercise for both myself and the pupils, in fact the whole class agreed that they had learned something from it. I am sure that the lesson is useful in furthering the pupils' understanding of place value in all number bases and it also gives the pupils the opportunity to develop their own mathematical ideas based on their past learning experiences.

Finally, if anyone can modify and improve any of the tests mentioned here I would be pleased to hear from you.

Mathematics in School, March 1985 7

This content downloaded from 5.198.113.170 on Tue, 22 Apr 2014 08:22:31 AMAll use subject to JSTOR Terms and Conditions