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Use Euclids division lemma to show that the cube of any positive integer is of the form 9m,9m+1or9m+8 |
Answer» ♡♥JAY SHREE KRISHNA♡YOUR♡ANSWER♡AS♡FOLLOW♡ ➡cube of any positive integer:- 9m, 9m+1, 9m+8 ➡Let a be any positive integer and b=3 0<equal to r < b 0<equal to r < 3 r= 0, 1, 2 a = 3q + r...............(1) ➡ If r= 0 in equation (1) a=3q+0 a=3q cube both side a³= 27q³ a³= 9(3q³) a³= 9m [m= (3q³)] ➡If r=1 in equation (1) a= 3q+1 cube both side a³= (3q+1)³ = 27q³+1+3(3q)²+(1)+3(3q)(1)² =27q³+27q²+9q+1 |___________| =9(3q³+3q²+q)+1 a³= 9m+1 [m= (3q³+3q²+q)] ➡If r=2 in equation (1) a=3q+2 cube both side a³= (3q+2)³ =27q³+8+3+(27q²)+2)+3(27q)(4) =27q³+54q²+36q+8 =9(3q³+6q²+4q)+8 a³=9n+8 [m= 3q³+6q²+4q)] ♡♥PLS MARK AS BRAINLIST FOR THIS BIG ANSWET ♡♥JAY SHREE KRISHNA
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