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Home / Elementary Number Theory / Show that the expression a(a^2+2)/3 is an integer for all a≥0.

Show that the expression a(a^2+2)/3 is an integer for all a≥0.

Problem: Show that the expression a(a2+2)/3 is an integer for all a≥0.

Solution: According to division algorithm, every a is of the form 3q, 3q+1, 3q+2.

Assume the first of this cases then a(a2+2)/3 = 3q{(3q)2+2}/3= 3q(9q2+2)/3= q(9q2+2).

Which clearly is an integer.

Similarly if a = 3q+1 then

a(a2+2)/3 = (3q+1){(3q+1)2+2}/3=(3q+1) (9q2+6q+1+2)/3= (3q+1) (9q2+6q+3)/3= (3q+1) (3q2+2q+1)

so a(a2+1)/3 is an integer in this also.

Finally, for a = 3q + 2, we see that

a(a2 + 1)/3 = (3q + 2){(3q + 2)2+ 2}/3 = (3q + 2)(9q2 + 12q +6)/3

=(3q + 2) (3q2 + 4q +2)

Consequently our result is established foe all cases.                                                                   (Proved)

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