The answer
(a) explained below
(b) \(k = 98\)
O-Level E-Math 2017 Paper 1 Question 18 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 18 of the O-Level E-Math 2017 Paper 1. It tests perfect square via even prime indices, in the Numbers (primes, powers, roots) area. It is worth 5 marks: 3 + 2. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a) Write each number in prime-power form: \(28 = 2^2 \times 7\) and \(63 = 3^2 \times 7\). Then \[28 \times 63 = 2^2 \times 3^2 \times 7^2 = (2 \times 3 \times 7)^2 = 42^2.\] Every prime appears with an even index, so the product is a perfect square (equal to \(42^2 = 1764\)).
(b) \(28 = 2^2 \times 7\). For \(28k\) to be a perfect cube, every prime index in \(28k\) must be a multiple of 3. The \(2^2\) needs one more factor of 2 (to make \(2^3\)) and the \(7^1\) needs two more factors of 7 (to make \(7^3\)), so \(k = 2^1 \times 7^2 = 2 \times 49 = 98\). Then \(28 \times 98 = 2744 = 14^3\). The smallest such \(k\) is \(98\).
Answer: (a) explained below
(b) \(k = 98\)
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a perfect square via even prime indices question from Numbers (primes, powers, roots), worth 5 marks: 3 + 2.
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