Question

If $n$ positive integers are taken at random and multiplied together, then the probability that the last digit of the product is $2,4,6$ or $8$

• A. ${\left(\frac{4}{5}\right)}^n$
• B. $\frac{3^n-2^n}{5^n}$
• C. $\frac{4^n-2^n}{5^n}$
• D. None of these

Answer 1

Answer: (A)

${\left(\frac{4}{5}\right)}^n$

The last digit of the product will be $1,2,3,4,6,7,8$ or $9$ if and only if each of the $n$ positive integers ends in any one of these digits.
The probability of it is $\frac{8}{10}=\frac{4}{5}$
Therefore, the probability that the last gigit of the product of $n$ integers is $1,2,3,4,6,7,8$ or $9$ is ${\left(\frac{4}{5}\right)}^n$