Question : If a 10-digit number 75462A97B6 is divisible by 72, then the value of $\sqrt{8 A-4 B}$ is:
Option 1: $\sqrt{30}$
Option 2: $\sqrt{27}$
Option 3: $\sqrt{21}$
Option 4: $\sqrt{28}$
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Correct Answer: $\sqrt{28}$
Solution :
A number is divisible by 72 if and only if it is divisible by both 8 and 9 (since 72 is the product of 8 and 9, and 8 and 9 are coprime).
1. For a number to be divisible by 8, its last three digits must form a number that is divisible by 8.
In this case, the last three digits are 7B6.
For 7B6 to be divisible by 8, "B" must be 3.
2. For a number to be divisible by 9, the sum of its digits must be divisible by 9.
If we substitute B = 3 into the number 75462A9736,
the sum of the digits = 7 + 5 + 4 + 6 + 2 + A + 9 + 7 + 3 + 6 = 49 + A
For this to be divisible by 9, A must be 5.
$\therefore\sqrt{8 A-4 B}=\sqrt{8 \times 5 - 4 \times 3} = \sqrt{28}$
Hence, the correct answer is $\sqrt{28}$.
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