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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