Question

Let $N$ be the least positive integer such that whenever a non-zero digit $c$ is written after the last digit of $N,$ the resulting number is divisible by $c.$ The sum of the digits of $N$ is

• A. $9$
• B. $18$
• C. $27$
• D. $36$

Answer 1

The correct option is A $9$

Let $N=a_0+10a_1+{10}^2{a}_2+\cdots$

Now, according to the question, $c$ is added at the last of $N.$
So, the new number is of the form $10N+c.$

Given that $10N+c$ is divisible by $c$
$\Rightarrow 10N$ is divisible by $c.$
$c$ can take any values from $\{1,2,\cdots ,9\}.$ So, for $10N$ to be divisible by $c,10N=(9\times 8\times 7\times 5)k$

$\Rightarrow N=252k$
So, the least value of $N=252$