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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?
Let the 6 digit number be _ _ _ 42_
It is divisible by 2,3,5,7,11,13
Since the number is divisible by both 2 and 5 the last digit of the number must be 0.
The number is also divisible by 3, 7, 11, and 13.
Hence the number must also be divisible by 7*11*13.
=7*11*13 = 1001.
A number which is a multiple of 1001 is of the form abcabc.
This is because abc*(1001) = abc*(1000+1) = abc000 + abc = abcabc.
Hence the number is 420420.
The first digit is 4.
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