Gus's magic numbers: A key to the divisibility test for primes

Gus's magic numbers: A key to the divisibility test for primesAuthor(s): CHARLENE OLIVERSource: The Arithmetic Teacher, Vol. 19, No. 3 (MARCH 1972), pp. 183-189Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187975 .

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Gus's magic numbers: A key to the divisibility test for primes

CHARLENE OLIVER

Continuing what she began as a graduate student working toward a master's degree in mathematics, Charlene Oliver is teaching mathematics classes for students majoring in elementary education at West Texas State University in Canyon, Texas. She has also taught mathematics and science at the high school level. Her son, Gus Oliver, Jr., was a sixth grader at South Middle School, Borger, Texas, when he discovered his "magic numbers."

J''' of us are familiar with a divisibility test for 3 in which we add the digits of the number being tested. Some of us are familiar with a divisibility test for 7. How many of us, though, are aware that a divisibility test for 7 may also be used for 3?

I was not aware of this myself until recently, when it was pointed out to me by my eleven- year-old son, Gus.

"The reason this is true," he said, "is because both 3 and 7 have the same magic number." Gus's "magic number," for him, was the key unlocking the door to a divisibility test

that will work for all prime numbers except 2 and 5. This is the story of his discovery. Gus first became interested in divisibility tests last year while preparing a project on

prime and composite numbers for the annual Borger Science Fair. At that time he formu- lated the usual divisibility rules for 2, 3, and 5.

Some time after the science fair, I demonstrated tests for divisibility by 7 and 1 1 to him. I did not explain why these two tests work. Therefore, I was surprised to discover later, in his notebook on prime and composite numbers, not only an explanation of the procedures used in making the tests but also an explanation of why they work.

For testing the divisibility by 7, he had written: To check for numbers divisible by 7 you cross out the last digit, double the crossed out

digit, and subtract it from the new number. Repeat until you reach a number less than 100 and if that number is divisible by 7, the original number is divisible by 7 and composite but if the new number isn't divisible by 7 then the original number isn't divisible by 7. The numbers you are really subtracting are multiples of 21 and 21 is a multiple of 7.

March 1972 183



Gus also had recorded the following calculations in his notebook:

Fig. 1

When I questioned Gus about this test, he explained, "The magic number here is 21. Just look what we can do to the multiples of 21: 0, 21, 42, 63, • • , 189." He illustrated as follows:

00 2/ 4/ -0(2 X 0) -2(2 X 1) -4(2 X 2)

0 0 0

6/ 18^ -6(2 X 3) • • • -^8(2 X 9)

0 0

"In 21, you see," he said, pointing, "if you cross out the last digit 1, multiply the crossed- out digit by 2, and subtract the product from the new number, you get zero - which is the same as subtracting 21.

"This means that when using the rule for 7, if the crossed-out digit is 1, you end up actually subtracting 21, or 21 X 1, if it is the first subtraction; 210, or 21 X 10, if it is the second subtraction; 2,100, or 21 X 100, if it is the third subtraction; and so on. If the crossed-out digit is 2, you end up subtracting 42, or 21 X 2; 420, or 21 X 20; 4,200, or 21 X 200; and so on.

"Since you are just subtracting multiples of 21, which are multiples of 7, then if the final number is a multiple of 7, the original number is a multiple of 7."

His explanation for divisibility by 11 is similar to the one for 7. In his notebook he recorded :

184 The Arithmetic Teacher

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У. 7/7fZ ла. J¿v**JfU Át, 7



Fig. 2

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Jfou /пш- /n^v¿a. nJifaciX AurilüL /y<Ku VdnrNL &иЛ мпХК О (Уь /m/M* O. Jý* rybuj Сыт (X/X млМ, О лХ m¿ ¿¿¿a/tmIAu ¿у> ¡I /nul xf' AX, ám /ЩХММР 0 M М/щ oC¿a/<ááJAX'

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of // lkaÏA/6~Lu mJMaoxU/ aAo Oj I Ij 12) 33, 4% 5$; M, 7?J ЪЪ, Я% НО, ?Щ

7t 770 Л CÍO ■Ц'*1) ~110(i''7o) -lí'<7) -Т?о(н*ъ) ô 000. 'I -ÏO0

%% a, JmmJÀc by lì CfZuMrti ЛЛМс Myli

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For divisibility by 11, Gus enlightened me further, 'the magic number is 11 itself. Look at the multiples of 1 1 . For example, you see that in 66 if you cross out the last digit 6, multiply the crossed-out digit by 1, and subtract the product from the new number, you get zero - which is the same as subtracting 66. Therefore, when using this rule, if the crossed- out digit is 6, you are actually subtracting 66, 660, 6,600, and so on. So again we are just subtracting multiples of the prime number."

After he finished explaining why the tests for divisibility by 7 and by 11 work, I said,

March 1972 185



"I wonder if there is a divisibility test for any other primes. For example, is there a test for 13? I don't recall ever seeing one."

His big brown eyes sparkled as he responded enthusiastically, "You just gave me a great idea! I'll try to find one."

For the next two days Gus spent long periods in his room, with the door closed. I later found discarded in his wastebasket several sheets of paper on which he had figured, using the number 13. On one of the discarded sheets he had written the following explanation for divisibility by 13:

Fig. 3

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me




Although we see that this method will always work, as would two other discarded methods that I found, Gus was not content with this method because, as he told me later, it took too long to test large numbers.

Finally, satisfied at last, Gus emerged from his room and showed me his notebook. He had written this final conclusion:

Fig. 4

,))M-fX¿JL'(ti' ¿yC 7/. JAM/ JjbMAj muUhhol A/JiM Mr&'Jb ̂6^1 all ÁA/sniť <Ob¿or>iAtA¿s Мр<<ф£ 7, *s7^d. S. ЗЛА, .ПЧл^улХ^СУЬ JJbcW

ая- ¿Hit* X^ju M*pli> ¿t/wji J(X unsoyxJ-y-

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ný^JirUÁ MjCLjU Л& ^гилЛХс^Ау /t^uc СЛШЫ &uX UooCt Jy*. &Я jAc JsbcÂ/ â£ Лки f^Oj^L Л^НО siMvU ¿uyvz>{

March 1972 187



His table follows. We see that his table, if not his description, clearly indicates that the magic number is the prime's first multiple whose last digit is 1 .

r

3 Д1 2?I=Z, Z

// // ^/* /, / 13 II 7-/'" % 1 17 51 fri* ¿ Г /^ /7/ /7->- /^ /7 ^3 /^/ /¿f/- /Í //

3/ 3/ ^-/r 3, 5

^7 /У/ /^/т /^ /У 53 37/ .W 37y i7 57 53 Í Л"/г/3 ^ í/ ó/ ¿*/-¿/ ^>

7/ 7/ 7--/s 7 7 73 5// ГЛУ'^ Г/ А 77 71/ 7/r/-7¿ 7/ • F3 Л/ Я-7'^ $$

V 2*11 2el*}-2% 11 ф /ÕI 101 10*)* IV, ¡O

Fig. 5




"Why will this test work for all other primes and not for 2 or 5?" I asked. "It's because 2 and 5 are the only primes that will go into 10, 100, 1,000, and so on, evenly,"

he answered. "In the first example for 7," he continued, "although the final result is actually 56,000, we may consider it as 56 because 56,000 will be divisible by 7 only if 56 is divisible by 7. The same is true for all other primes except 2 and 5."

Gus then pointed out that since 3 and 7 have the same magic number, they may both be tested at the same time. For example, from the final result 56 in the first problem on divisibility by 7, we may conclude not only that 71,792 is divisible by 7, since 56 is divisible by 7, but also that 71,792 is not divisible by 3, since 56 is not divisible by 3.

He went on to say that 3 and 7 are not the only pair that may be tested at the same time. Any pair of primes whose product ends in the digit 1 may be tested at the same time by using the product as the magic number. Examples are the pairs: 3 and 7; 7 and 13; 13 and 17; 11 and 31; and 19 and 29.

Summarizing Gus's explanations, the general method for testing divisibility by the primes 3,7, 11, 13, ••• is this:

1. Choose a prime number and a natural Let us test 1,292 for divisibility by 17. number on which to test the divisibility by the prime number.

2. Find the "magic number." This will be Multiples of 17 are 0, 17, 34, 5J_, • • ; the prime's first multiple whose last digit the "magic number" is 51. is 1.

3. Divide the last digit of the "magic The multiplier is 5 -5- 1 = 5. number" into the number named by the remaining digits to obtain the multiplier.

4. Cross out the last digit of the number 129/ being tested, multiply the crossed-out -10 (5 X 2)

digit by the multiplier, and subtract the 119 product from the new number.

5. Continue as in step 4 until the final result 1 1# is small enough to determine divisibility -45 (5 X 9) easily. - 34

6. Conclude that the original number is Since -34 is divisible by 17, we conclude divisible by the prime if the final result that 1,292 is divisible by 17. is divisible by the prime; otherwise it is not.

This story, I believe, confirms that divisibility tests for 7 and 11 could be used as enrich- ment material and introduced along with the simpler divisibility tests during the study of

prime and composite numbers at about the fifth- or sixth-grade level. Students could then be given an opportunity to find how these tests work, to discover that

the test for 11 is just a variation of the one for 7, and possibly to discover, as sixth-grader Gus did, that the test for 7 will work for every prime except 2 or 5 simply by changing the multiplier, which is derived from the prime's own magic number.

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March 1972 189



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Gus's magic numbers: A key to the divisibility test for primes