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The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

# 2-digit Square

##### Age 14 to 16 Challenge Level:
Thank you Abdul for this solution.

The 2-digit number is either $65$ or $56$.

### Proof:

Any 2-digit number can be represented as $10a + b$. We need $(10a+b)^2 - (10b+a)^2 = 99a^2 - 99b^2 =9 \times 11 \times (a^2 - b^2)$ to be a square.

This means that $(a^2 - b^2)$ must be 11 and so $(a - b)(a + b) = 11$ making, $a - b = 1$ and $a + b = 11$. This gives $a = 6$, and $b = 5$.

If we find a solution with $a > b$ then, by reversing the digits, we get a second solution.