50 BOOK OF MATHEMATICAL PROBLEMS. 



221. Prove that 



XY. Properties of Numbers. 



222. If n be a positive whole number, 



2 2n + I5n- 1 is divisible by 9, 



(27j + l) 8 -2rc-l ............... 120, 



3"+'_8n-9 ............... 64, 



3 2n+8 + 40w-27 ............... 64, 



- 56^-243 ............... 512. 



223. If 2p+ 1 be a prime number (|p) 8 + (-l) p is divisible 

 by 2p+l. 



224. If p be a prime number, r _ l C n + (- I)"" 1 is divisible by p. 



225. If 7i 1, n + 1 be both prime numbers > 5, n must be 

 of one of the forms 30tf, or 3012j and n 2 (n*+16) will be 

 divisible by 720. 



226. If 7i '2, Ti + 2 be both prime numbers > 5, n must be 

 of one of the forms 30* + 15 or 30$ 9. 



