Question

Find the odd term among: 23, 35, 57, 711, 1113, 1316.
$\begin{align}
  & a)23 \\
 & b)35 \\
 & c)57 \\
 & d)1316 \\
\end{align}$

Answer 1

Hint: Note that the last term on the previous number is the same as the first term of the next number. Now we will split each term into two parts and note the similarity in those two. Hence we will get a sequence and we will check which term does not fit the sequence.

Complete step-by-step solution:
Now before we start let us understand prime numbers and composite numbers.
Now any natural number can be grouped into either prime or composite numbers.
Now prime numbers are those numbers, which have only two factors that is 1 and the number itself. Hence all prime numbers cannot be divided by any number other than 1 or the number itself. For example $2, 5, 7, 11.$ Now all these numbers cannot be divided by any numbers except 1 and the number itself. Which means the only possible factorization of the numbers are $2 = 2 \times 1, 5 = 5 \times 1, 7 = 7 \times 1, 11 = 11 \times 1. $
Now composite numbers are the numbers, which are not prime numbers. Hence this means they will always have a factor other than 1 and number itself. These numbers can always be shown as the multiplication of prime factor and this is called prime factorization. For example, consider number 6. We have $6 = 3 \times 2$. Hence we can show 6 as the multiplication of prime number. Hence all such numbers are called composite numbers.
Now let us list the first 10 prime numbers first.
$1, 3, 5, 7, 11, 13, 17, 19, 23, 29.$
Now consider the given terms $23, 35, 57, 711, 1113, 1316. $
First, we note that the last number in the previous term is the first number in the next term.
Now we will split each number.
Let us split the first term 23 as 2 and 3. Now we have 2 and 3 are consecutive prime numbers.
Now again we will split 35 as 3 and 5. Hence we again have 3 and 5 as consecutive prime numbers
Now we will write 57 as 5 and 7, 711 as 7 and 11, 1113 as 11 and 13.
Here we can see that all numbers are made by writing consecutive prime numbers together.
Hence the next number should be 1315 as 13 and 15 are consecutive prime terms.
Hence we get 1316 does not fit in the given sequence.
Option d is the correct option.

Note: Note that all even numbers except 2 are composite numbers since all even numbers are divisible by 2. 2 is the only even prime number but all odd numbers need not be prime. For example $9 = 3 \times 3.$