0<}<2 RULES FOR ASCERTAINING SQUARE NUMBERS. 



which can never be squares, and thus at least preventing 

 them the trouble of perhaps many useless extractions. 

 Should you, Sir, deem these propositions of sufficient im- 

 portance to occupy a place in your useful nliseellany, your 

 insertion of them will oblige, 



Sir, your very humble servant, 



W. SAINT. 



yVqiftrcnum- Proposition 1. — A square number cannot terminate with 



l# cannot ter- - - >' _ 



minatewithS, *> J > n op b# 



S, 7, or 8: Demonstration. — The terminating figure of every product 



arises from the multiplication of the terminating figures of 

 its factors. The terminating figure therefore of every square 

 number must arise from the product of 0X0, IX 1, 2X2, 

 3X3, 4X4, 5X5, 6xt>, 7X7, 8X8, 9X9; and these 

 products it is evident, can only end with 0, 1, 4, 9, 6, and 5, 

 and never therefore with 2, 3, 7, or 8. Q. E. D. 

 or with an odd Prop. 2. A square number cannot terminate with an odd 



Demonstration, Since every square number ending with 

 must have its root ending with 0, such a root must be of 

 the form 10 m y and the square number itself therefore of 

 the form 100 m* \ where it is evident, that whatever value 

 be given to m, the product 100 X m a must terminate with 

 two ciphers! It is also obvious, that it cannot end with 

 more than two, unless m a end with an 0; which again can 

 only be when m is of the form 10 m, or rri 1 of the form 100 w~, 

 and therefore 100 X tn % of the form 10000 X « 4 , which must 

 end in at least four ciphers; and so on as far as we please. 

 Q. E. D. 



If it fonrvnate Prop. 3. If a square number terminate with 4, the last 

 v.i'li i, thelavt fjo Ure but one will be an even number. 



but one °_ . .. „ ., , , 



mtit be even. Demonstration, 1 or swell a square number must have i+s 



root ending in 2, or 8 ; this root will therefore be of the 

 . fgrm 10 ?n 4- 2, and its square of the form 100 m 1 +- 40 m 

 \ 4 ; where, whatever value be given to m, the sum or dif- 

 ference of the first and second terms will give an even num- 

 ber of tens, and an even number of tens plus 4 must have 

 the last figure but one an eveti number. 



' Prop, 



