The Properties of the Figure Nine.

Auckland Star, Volume XXI, Issue 4124, 22 September 1883, Page 6 (Supplement)

The Properties of the Figure Nine.

(To the Editor). Sin,—l send you tho solutions of the two questions on this subject which appeared in your Sui'l'Lkmknt of September Sth. Ist. The first (correcting a trifling inaccuracy in the statement) is as follows:—" If the sum of the digits of any number bo subtracted from that number, the remainder will bo exactly divisible by 9." Write a number, which v.'o may supposo to consist of three digits, in the manner I pointed out in my former letter, we get lOOtu+lO.r+l/ subtracting w+ x+y wo get 90i«+ 9n •which, of course, is oxactly divisiblo by 9. Further, this remainder being a multiplo of 9, the sum of its digits (as I showed in my last letter) must be either 9 or a multiple of 9 which explains tho rest of the question. 2nd. Tho second problem, upon investigation, resolves itself into the following simple proposition:the truth of which, I think, fow will deny. For let a and b bo the two numbers, each consisting of two digits, and let a bo the greater. Following the steps indicated, wo get g9 _ 6 add tho greater 90+ a-& Hero yro have 99 + tho difference of the two numbers. Now erase left-hand digit which from the nature of the question, must be 1; or, in other words, subtract 100, wo have 9a _ l oo+a-& add 1 99+1-100+«-b or the difference botwecn tho numbers. I need not trouble to send you solutions of tho questions which appeared in your last issue? a* ono of them is identically tho same m the first of the above, while tho truth of the other is at onceapparcnt upon applying the same, nothod of «*&W% in this and my former letter.-I am, Stc, TonuNGA. To Awamutu.