A test to check divisibility by any numberAuthor(s): Ruth BrownSource: The Arithmetic Teacher, Vol. 12, No. 6 (OCTOBER 1965), p. 459Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186970 .

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The Euclidean algorism is typical of most methods of computing in that its use proves to be especially efficient with some examples and less efficient than other algorisms with other examples. The num- ber line development, however, provides a basis for understanding this algorism, which may prove logical to some students who have difficulty understanding the more commonly taught procedures for finding the greatest common factor.

The Euclidean algorism can, of course,

be used in determining the least common multiple (LCM) of two numbers. The LCM can be found by dividing one of the two numbers by the greatest common factor, and then multiplying the resulting quotient times the other "original" num- ber. For elementary school arithmetic, however, the greatest value of the Euclid- ean algorism seems to lie in testing the re- ducibility to fractions which involve large numerals and in simplifying the terms of such fractions.

A test to check divisibility by any number Choose a number - to it add a multiple of seven so that the last digit of the sum will be zero. Next add a multiple of seven to the sum so that the last two digits will be zero. Keep repeating this process adding one more zero each time, until the number can be easily divided by seven. If the remaining number is divisible by seven then the original number was a multiple of seven.

Example: 98,763,429,018 98,772,000,000

42 98,000,000

98 , 763 , 429 , 060 98 , 870,000,000 840 630,000,000

98 , 763 , 429 , 900 99 , 500,000 , 000 9,100 3,500,000,000

98,763,439,000 103,000,000,000 91,000 77,000,000,000

98 , 763 , 530 , 000 180 , 000 , 000 , 000 770,000

98,764,300,000 7,700,000

98,772,000,000 180,000,000,000 is not divisible by seven, there- fore, the original number was not either.

Example: 425,691,105 425,740,000

35 560,000

425,691,140 426,300,000 560 7,700,000

425,691,700 434,000,000 6,300 56,000,000

425 , 698 , 000 490 , 000 , 000 42,000

425,740,000

Since 490,000,000 is divisible by seven so was the original number. This same method can be used to discover if a number a is divisible by another number b by simply adding multiples of ò to a, instead of multiples of seven. - Becky Lemmon, 8th Grade, McKinstry Junior High, '63, Waterloo, Iowa.

(A note from her teacher.) The students in my eighth-grade accelerated algebra class were interested in factors and divisibility. They had studied the following theorems:

1 "For positive integers а, b, and c, if a is a fac- tor of both b and c, then a is a factor of (b+c)."

2 "For positive integers a, b, and c, if a is a fac- tor of b, and a is not a factor of (6 -f-c) then a is not a factor of c."

3 "For positive integers a, b, and c, if a is a factor of by and о is a factor of (6 +c) then a is a factor of c."

The class had reviewed the articles, "Divisibility by Odd Numbers" in The Arithmetic Teacher, March 1960, "More on Divisibility by Seven and Thirteen" from The Arithmetic Teacher, April 1961, and "A Test for Divisi- bility" from The Mathematics Teacher, December 1960. But, Becky was not satisfied. She thought all these were interesting but not too valuable to her. The work above is what she brought to class one day expressed in her own words. - Ruth Brown, McKinstry Jr. High School, Waterloo, Iowa.

October 1965 459

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