Find the largest number which divides $2053$ and $967$, leaving a remainder of $5$ and $7$ respectively.
Step-by-Step Solution
Key Concept: Required number is $\text{HCF}(2053 - 5, 967 - 7) = \text{HCF}(2048, 960)$.
Stepwise Solution:
Numbers to be divided completely: $2053 - 5 = 2048$ and $967 - 7 = 960$. [0.5 Mark]
Prime factorisation: $2048 = 2^{11}$, $960 = 2^6 \times 3 \times 5$. [1.0 Mark]
$\text{HCF}(2048, 960) = 2^6 = 64$. Required largest number is $64$. [0.5 Mark]
Marking Scheme:
• Subtracting remainders: 0.5 Mark
• Prime factorisation of 2048 and 960: 1.0 Mark
• Evaluating HCF $= 64$: 0.5 Mark
Correct Answer: