Discussion

Question:

If a, b and c are 3 consecutive integers between –10 to +10 (both inclusive), how many integer values are possible for the expression $\frac{a^{3} + b^{3} + c^{3} + 3 a b c}{{(a + b + c)}^{2}}$ ?

Explanation:

If a, b and c are three consecutive numbers
⇒ a = b – 1 and c = b + 1

Substituting this in the expression we get the expression we get,

= $\frac{{(b - 1)}^{3} + b^{3} + {(b + 1)}^{3} + 3 (b - 1) b (b + 1)}{{(b - 1 + b + b + 1)}^{2}}$

= $\frac{6 b^{3} + 3 b}{{(3 b)}^{2}}$

= $\frac{2 b^{2} + 1}{3 b}$

Only for b = ±1 is the above expression an integer.

Hence, option (c).

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