Question

Find a 4-digit odd number using each digit 1, 2, 4 and 5 only once such that when the first and the last digits are interchanged, it is divisible by 4.

Answer 1

Hint: To solve the given problem we need to know the divisibility test for $4$. Prior to it the first step is to find the odd numbers that could be found with the given numbers. The odd number is formed when the one placed in the number formed is an odd number which means is either of $1$ or $5$ according to this problem.

Complete step by step answer:
The question asks us to find the number which is divisible by $4$ on interchanging the first and the last digits of the number formed, when these four numbers are given which are 1, 2, 3 and 5. The number is formed with the given four numbers which should not be repeated.
To form 4-digit odd numbers the ones placed in the number should be an odd number. Since in this question we are asked to make an odd number, where we only have two odd digits which are $1$ and $5$. So the odd the thus formed are:
$1245$, $1425$, $5241$…….. and many others . Total number of odd numbers formed from these four digits is $12$.
The second condition given in the question is that on interchanging the first and the last digit the number becomes divisible by $4$. For a number to be divisible by $4$ the last two digits of the number should be divisible by $4$. On keeping this fact in mind we come to know that $24$ and $52$ are the numbers if put in the last will be divisible by $4$ get:
$1524$,$5124$,$1452$ and $4152$
Now, we would check that which number fulfills both the conditions that is the number should be odd and on interchanging the last and the first digits the number become divisible by $4$. The numbers which are divisible by $4$ after considering conditions are $1524$ and $1452$ .
$\therefore $ $4521$ and $2451$ are the 4-digit number using each digits 1, 2, 4 and 5 only once such that when the first and the last digits are interchanged, it is divisible by 4.

Note: To solve this type of question we need to consider the conditions properly. The first condition needs to be solved first and then the second condition should be worked upon. The divisibility criteria should be known to us, like for a number to be divisible by $4$ the number formed by the last two digits should be divisible by $4$.