The product of the consecutive integers is a multiple of 210, find the difference between highest and smallest value of the middle integer?

Question

In the video titled problem questions 2 in number system, it is mentioned there is only one solution which is 5,6,7. However 13,14,15 can also be a solution.

Divakshi Arora · Accepted Answer

The question asks for three consecutive integers whose product is a multiple of 210. Some answers say the only solution is 5, 6, 7, but that&rsquo;s not true. 210 = 2 &times; 3 &times; 5 &times; 7, so any set of three consecutive numbers where the product contains all those prime factors will work.
13, 14, 15 is another valid set. 13 &times; 14 &times; 15 = 2730, which is divisible by 210. The middle integer there is 14, and in 5, 6, 7 the middle is 6. So, the difference between the highest and smallest middle values is 14 &ndash; 6 = 8.
The catch is, the question isn&rsquo;t asking for the middle integer itself, it wants the difference between the highest and smallest possible middle integers. So the final answer should be 8, not just the middle integer of one set.