A prime number, \(x\), is greater than \(75\) but less than \(81\).
State the value of \(x\).[1]
It is given that \(x\) can be expressed as a product of \(h-22\) and \(h+56\), where \(h\) is a positive integer.
Find the value of \(h\), showing your working clearly.[2]
Express \(127\,008\) as a product of its prime factors.[1]
Hence, find the largest possible integer value of \(\sqrt[3]{\frac{127\,008}{2k}}\) and the corresponding value of \(k\), where \(k\) is a positive integer.[3]
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Answer:(a)(i) \(79\) (ii) \(h=23\) (b)(i) \(127\,008=2^5\times3^4\times7^2\) (ii) largest value \(=6\), \(k=294\)