tmua.me Tau Paper 2 · Question 7

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tmua.me Tau Paper 2 · Q7

Paper 2Number & countingLogic & proof

For each choice of integers $r,s,t,u$ satisfying

0\le r,s,t,u\le12,

put $N=2^r3^s5^t7^u$ . Let $D(N)$ be the largest positive integer with the following property: for all integers $a,b$ ,

N\mid a^2+ab+b^2\quad\Longrightarrow\quad D(N)\mid ab(a+b).

How many of these distinct values of $N$ have $D(N)$ a perfect square?

3822

4116

4459

3276

3528

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