Let $k$ be the number of 10 digit numbers, formed using the digits 0,1,2...9 ( with repetition allowed ) ,such that the product of any two consecutive digits in the number is a prime number ,then which of the following options are correct?

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$k$ is a perfect square of a natural number

$\frac{k}{2}$ is a perfect square of a natural number

$2 k$ is a perfect cube of a natural number

$4 k$ is a perfect cube of a natural number

answer is B, C.

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Detailed Solution

clearly digit 1 should occur at alternate places of the number and at the remaining 5 places either 2,3,5 or 7 should appear. If number starts with 1, then we have $4^{5}$ such numbers and when number starts with 2,3,5 or 7 number of such numbers = $4^{5}$ so total number of required numbers= $2 \times 4^{5}$

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