ALGEBKA. 51 



7. If n be a whole number, n+ 1 and n* n + \ cannot 

 both le square numbers. 



\ The whole number next greater than (3 + J5)* is divi- 

 sible by 2". 



'. The integral part of -j= (J3 + J5)*- 1 is divisible by 2". 



230. The equation 8 -2y 8 = fcl cannot be satisfied by any 

 integral values of x and y different from unity. 



231. The sum of the squares of all the numbers less than a 

 given number N and prime to it is 



and the sum of the cubes is 



I he different prime factors of N. 



XVI.. Probabilities. 



A and B throw for a certain stake, each one throw N 

 10. A'* die i* marked 2, 3, 4, 5, G, 7 and Jfs 1, 2, 3, 4, 5, 6 ; 

 and equal throws divide the stake. Prove that A' a expectation is 



the stake. What will A'* expectation be if equal throws go 



>. A certain sum of money is to be given to the one of 

 three persons A, B, C who first throws lu \M h three dice; sup- 

 ' them to throw in the tin- event L; 



(s \ f / 7 \ f 



ia j , and (Ts ( T-jrJ . 



4- 



