Question : A four-digit pin, say abcd, of a lock has different non-zero digits. The digits satisfy b =
2a, c = 2b, d = 2c. The pin is divisible by __________.
Option 1: 2, 3, and 5
Option 2: 2, 3, and 7
Option 3: 2, 3, and 13
Option 4: 2, 3, and 11
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Correct Answer: 2, 3, and 13
Solution :
Given, $b = 2a, c = 2b, d = 2c$
Finding all the values in terms of $a$.
⇒ $c = 2b = 2\times 2a = 4a$ ..............(1)
Similarly, $d = 2c = 2\times 4a = 8a$ ..............(2)
Now the pin is $abcd$,
⇒ $abcd = 1000a + 100b + 10c + d$
⇒ $abcd = 1000a + 100\times 2a + 10\times 4a + 8a$
$= 1000a + 200a + 40a +8a$
⇒ $abcd=1248a$
⇒ $abcd = (2\times 2\times 2\times2\times2\times 3\times 13)$
$\therefore$ the pin abcd is divisible by 2, 3, and 13.
Hence, the correct answer is 2, 3, and 13.
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