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# Integer Root (Tomato Subjective 43)

Show that the equation $$x^3 + 7x – 14(n^2 +1) = 0$$ has no integral root for any integer n.

Solution:

We note that $$14(n^2 +1) – 7x = x^3$$ implies $$x^3$$ is divisible by 7. This implies x is divisible by 7 (as 7 is a prime number). Suppose x= 7x’. Hence we can rewrite the given equation as:

$$7^3 x’^3 + 7 \times 7 x’ – 14 (n^2 +1 ) = 0$$.

August 28, 2013

### 1 comment

1. ISI B. Math solved questions needed