Question

Using the digits $4,5,6,7,8$, each once, construct a $5-digit$ number which is divisible by $132$

Answer 1

To check for divisibility by 132

First we factorize 132 as $132=4\ast 3\ast 11$

Now, if the Number obtained is to be divided by 132 then it has to follow $DivisibilityRules$ of 4,3 and 11 which are given below

$\left(i\right)$ Divisibility Rule of $4$ is the last $2$ digits of the number should be divisible by $4$

$\left(ii\right)$ Divisibility Rule of $3$ is the Sum of digits must be Divisible by $3$

$\left(iii\right)$ Divisibility Rule of $11$ is Take the alternating sum of the digits in the number, read from $left$ to $right$. ... So, for instance, $2728$ has alternating sum of digits $2-7+2-8=-11.$ Since -11 is divisible by 11, so is $2728$.

Now, Out of the Given Five Digits only $4,8$ are satisfying $\left(i\right)$ so last $2$ digits are either $84$ or $48$

Now $\left(ii\right)$ is also satisfying as Sum of digits $=30$

Now, we have to check $\left(iii\right)$

By $Hit$ and $Trial$ method we get

$\Rightarrow$ $67584$ and $57684$ (Also the sum of these digits is $0$ as it can be either multiple of $11$ or is $0$ )