Question

The number of 10-digit numbers such that the product of any two consecutive digits in the numbers is prime number, is

• A. 1024
• B. 2048
• C. 512
• D. 64

Answer 1

Answer: (B)

2048

The product of any two consecutive digits is a prime number only when one of them is $1$ and the other is a prime number.

So basically, every other number has to be $1$. The other digits have to be $2,3,5$ or $7$.

As we are talking about $10$ digit numbers,

Case $1$:

Consider numbers of the form $1x1x1x1x1x$, where $x=2,3,5$ or $7$.

For each x you have $4$ possible selections $(2,3,5,7)$ - so you multiply $4,5$ times(for the $5$ slots)

Possible numbers$=4\times 4\times 4\times 4\times 4\times 4^5=2^{10}$

Case $2$:

Consider numbers of the form $x1x1x1x1x1$.

For each x you have $4$ possible selections $(2,3,5,7)$ - you multiply $4,5$ times(for the $5$ slots)

Possible numbers$=4\times 4\times 4\times 4\times 4=4^5=2^{10}$

Thus, total possible numbers$=2^{10}+2^{10}=2\times 2^{10}=2^{11}=2048$.