Discussion
When 2629, 3481 and 4972 are divided by a two-digit number \(x\), then the remainder in each case is \(y\). What is the value of \((2x - 5y)\) ?
Answer: A
When different numbers leave the same remainder y on division by x, x divides the differences between the numbers. The differences are:
\[ 3481 - 2629 = 852 = 2^2 \times 3 \times 71 \]
\[ 4972 - 3481 = 1491 = 3 \times 7 \times 71 \]
The common factors are 3 and 71 (their product, 213, is the HCF). The two-digit divisor is therefore x = 71. The remainder is 2629 = 71 x 37 + 2, so y = 2.
\[ 2x - 5y = 142 - 10 = 132 \]
The options 133, 131 and 129 differ by one or three from 132, and come from a wrong remainder.
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