Zahir and Raman are at the entrance of a dark cave. To enter this cave, they need to open a number lock. Raman sees a note on a rock: “ ... chest of pure diamonds kept for the smart one ... number has six digits ... second last digit is 2, third last is 4 ... divisible by all prime numbers less than 15 ...”. Excited, Zahir and Raman seek your help: which of these can be the first digit of the six digit number that will help them open the lock?
Explanation:
Let the 6 digit number be _ _ _ 42_ It is divisible by 2,3,5,7,11,13
Hence, the number is a multiple of 2 × 3 × 5 × 7 × 11 × 13 = 30030
Now, 30030 × k = _ _ _ 4 2 _
Last digit would definitely be 0.
Second last digits which is 2 should be the last digit of 3 × k. Hence, last digit of k should be 4. ∴ k = 4 or 14 or 24 and so on.
The only possible value of k from the one's given above satisying the conditions given is 14.
∴ 30030 × 14 = 420420
∴ The first digit is 4.
Hence, option (e).
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