Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!linus!decvax!harpo!gummo!whuxlb!pyuxll!eisx!npoiv!npois!hogpc!houxm!ihnp4!ixn5c!inuxc!pur-ee!ecn-ec:ecn-pc:ecn-ed:vu From: ecn-ec:ecn-pc:ecn-ed:vu@pur-ee.UUCP Newsgroups: net.math Subject: Re: interesting division by 3 property Message-ID: <175@ecn-ed.UUCP> Date: Wed, 24-Aug-83 15:32:14 EDT Article-I.D.: ecn-ed.175 Posted: Wed Aug 24 15:32:14 1983 Date-Received: Fri, 26-Aug-83 02:22:58 EDT References: ihuxf.657 Lines: 38 This is the complete bag of tricks in divisibility I learned in 6th grade: + A number N is divisible by 3 or 9 if and only if the sum of the digits of N is divisible by 3 or 9 respectively + An even number divisible by 3 is divisible by 6 (trivial) + A number N is divisible by 11 if the sum of its 1st,3rd,5th,... digits is equal to the sum of its 2nd,4th,6th,... digits. This demands a proof: (not quite, but good enough to be generalize) abcdefg x 11 ------- abcdefg abcdefg -------- (bunch of numbers) add "or differ by a multiple of 11" after "equal to" + A number N is divisible by 4 or 8 if its last 2 or 3 digits resp. form a number divisible by 4 or 8 resp. (Since 4|100 and 8|1000 where | means divides) + A number is divisible by 5 or 10 if its last digit is 5 or 0 (for 5) or 0 (for 10) Combining them all together, you get a lot of composite number. I have not yet found anything about divisibility for primes other than 2, 3, 5, 11 In an article in Math. Monthly, one of its editor told the story about a suggestion for divibility for 7: Write N in base 8 (by the way, we have been talking only about base 10) then add the digits of N (base 8). If the sum (in base 8) is divisible by 7 then N is divisible by 7. Example: 21 = 25(base 8) whose sum is 7(base 8). One wonders who is going to do the base conversion !!! In fact, in base n, a number N is divisible by n-1 if and only if the sum of N's digit is divisible by n-1 (in base n). Hao-Nhien Vu (pur-ee!vu, changing into pur-ee!norris)