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 is/are  correct? 

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 $=2\times 4^5$