A prime number, \(x\), is greater than \(81\) but less than \(85\).
State the value of \(x\).[1]
It is given that \(x\) can be expressed as a product of \(h-24\) and \(h+58\), where \(h\) is a positive integer.
Find the value of \(h\), showing your working clearly.[2]
Express \(180\,000\) as a product of its prime factors.[1]
Hence, find the largest possible integer value of \(\sqrt[3]{\frac{180\,000}{2k}}\) and the corresponding value of \(k\), where \(k\) is a positive integer.[3]
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Answer:(a)(i) \(83\) (ii) \(h=25\) (b)(i) \(180\,000=2^5\times3^2\times5^4\) (ii) largest value \(=10\), \(k=90\)