Question

Write seven consecutive composite numbers less than $100$ so that there is no prime number between them.

Answer 1

Hint- This question can be solved by writing all composite numbers on the higher side i.e. from $90$ onwards.

In order to write seven consecutive composite numbers between $1$ and $100$ we have to consider the numbers of the higher side in counting because if we take smaller numbers there is more probability of prime numbers.
For example: If we start from first prime number i.e.
$2$ and $3$ are prime numbers followed by $5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97$.
When we see these numbers we get, composite numbers
$$\begin{gathered} 4,6,8,9,10,12,14,15,16,18,20,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42,44,45,46,48,49,50, \\ 51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,72,74,75,76,77,78,80,81,82,84,85,86,87,88,90, \\ 91,92,93,94,95,96,98,99. \end{gathered}$$
On seeing these numbers we find that we are not able to get $5$ consecutive composite numbers from $4$ to $88$ . And when we start from $90$ to $96$ we get seven consecutive composite numbers.

Note- Whenever we face such types of questions the key concept is that we have to locate composite numbers in the top series i.e. $90$ onwards . Like we did in this question. We can’t find consecutive seven composite numbers from $4$ to $88$.