The answer
(a) \(126 = 2 \times 3^2 \times 7\)
(b) \(k = 14\)
(c) \(x = 231\)
O-Level E-Math 2018 Paper 1 Question 20 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 20 of the O-Level E-Math 2018 Paper 1. It tests prime factorisation, in the Numbers (primes, HCF, perfect squares) area. It is worth 4 marks: 1 + 1 + 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) \(126 = 2 \times 63 = 2 \times 3^2 \times 7\).
(b) For a perfect square every prime index must be even. In \(2^1 \times 3^2 \times 7^1\) the indices of \(2\) and \(7\) are odd, so multiply by \(2 \times 7 = 14\) to make \(126 \times 14 = 2^2 \times 3^2 \times 7^2 = 1764 = 42^2\). The smallest \(k\) is \(14\).
(c) \(\gcd(x, 126) = 21 = 3 \times 7\) means \(x\) is a multiple of \(21\), but \(x\) must not share the extra factor \(2\) or a second \(3\) with \(126\) (else the HCF would exceed 21). Multiples of 21 between 200 and 300 are \(210, 231, 252, 273, 294\). Test from the smallest: \(210 = 2 \times 3 \times 5 \times 7\) has \(\gcd(210,126) = 42\) (rejected); \(231 = 3 \times 7 \times 11\) has \(\gcd(231,126) = 21\). So the smallest possible value is \(x = 231\).
Answer: (a) \(126 = 2 \times 3^2 \times 7\)
(b) \(k = 14\)
(c) \(x = 231\)
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 prime factorisation question from Numbers (primes, HCF, perfect squares), worth 4 marks: 1 + 1 + 2.
Yes. IP (Integrated Programme) schools teach the same O-Level Mathematics content; they just sequence it differently and set their own internal exams, so these worked solutions apply to IP students too.
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