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show that the square of any positive integer of the form 5 Cube + 1 is of the same form when Q is some integer |
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Answer» Let x be any positive integerThen x = 5q or x = 5q+1 or x = 5q+4 for integer x.If x = 5q, x^2= (5q)^2= 25q^2= 5(5q^2) = 5n (where n = 5q^2)If x = 5q+1, x^2= (5q+1)^2= 25q^2+10q+1 = 5(5q^2+2q)+1 = 5n+1 (where n = 5q^2+2q )If x = 5q+4, x^2= (5q+4)^2= 25q^2+40q+16 = 5(5q^2+ 8q + 3)+ 1 = 5n+1 (where n = 5q^2+8q+3 )∴in each of three cases x^2is either of the form 5q or 5q+1or 5q+4 and for integer q. |
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