Discussion
The greatest number that divides 87 and 123 leaving respectively 7 and 3 as remainders is :
Answer: C
The number is 40.
If the number leaves remainder 7 on 87 and remainder 3 on 123, it divides \(87 - 7\) and \(123 - 3\) exactly, so it is the HCF of those two numbers.
\[ \begin{aligned} 87 - 7 &= 80 = 2^{4} \times 5 \\ 123 - 3 &= 120 = 2^{3} \times 3 \times 5 \\ \text{HCF} &= 2^{3} \times 5 = 40 \end{aligned} \]
Check: \(87 = 2 \times 40 + 7\) and \(123 = 3 \times 40 + 3\). The divisor must also be larger than both remainders, which 40 is. 20 also divides 80 and 120 but is not the greatest such number, 80 does not divide 120, and 36 divides neither.
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