Higher June 2022 Paper 3 Q26
26 \(Q\) and \(R\) are two numbers.
As a product of prime factors,
\(Q = 2^3 \times 3 \times a^3\)
\(R = 2^4 \times 3^2 \times a^2\)
(a) The highest common factor (HCF) of \(Q\) and \(R\) is 4056
Work out the value of \(a\). [2 marks]
(b) Work out the lowest common multiple (LCM) of \(Q\) and \(R\). [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(2^3 \times 3 \times a^2\) or \(24a^2\) (= 4056) or (\(a^2\) =) \(\dfrac{4056}{2^3 \times 3}\) or (\(a^2\) =) 169 or \(\sqrt{169}\) | M1 | oe eg \(8 \times 3 \times a^2\) |
| 13 | A1 |
Additional guidance
| Condone \(a^2 \times 24\) for M1 | |
| Fully correct prime factor decomposition with values 2, 2, 2, 3, 13, 13 shown without 13 chosen as the final answer | M1A0 |
| Embedded answer \(2^3 \times 3 \times 13^2\) | M1A0 |
| \(\pm 13\) or \(-13\) | M1A0 |
| \(4056 \div 2^3 \times 3\) unless recovered to 169 | M0A0 |
| Answer | Mark | Comments |
|---|---|---|
| \(2^4 \times 3^2 \times a^3\) or \(144a^3\) or \(2^4 \times 3^2 \times (\text{their } 13)^3\) or \(13 \times 4056 \times 2 \times 3\) or \(52728 \times 6\) or \(24336 \times 13\) | M1 | oe eg \(144 \times (\text{their } 13)^3\) \(16 \times 9 \times 2197\) |
| 316 368 | A1ft | ft their 13, which must be an integer \(\gt 13\) |
Additional guidance
| eg 14 on answer line in part (a) can follow through to \(144 \times 14^3 = 395\,136\) | M1A1ft |