The product of three consecutive positive integers is a multiple of 210. What is the difference between the highest and the lowest possible value of the middle integer?

Question

the answer given in the video is 0, but 35,36,37 and 105,106,107 are also factors of it. how did you arrive at 0 then?

Divakshi Arora · Accepted Answer

For the product of three consecutive positive integers to be a multiple of 210, those three numbers together must include the prime factors of 210: 210 = 2 &times; 3 &times; 5 &times; 7.
That means, among the three consecutive numbers, their combined product must be divisible by 2, 3, 5, and 7. So we need to find sets of three consecutive numbers where all these primes are covered.
Try multiples of 105 (LCM of 2,3,5,7 = 210):

35 &times; 36 &times; 37 = 46620, divisible by 210
105 &times; 106 &times; 107 = 1192380, also divisible by 210

Both satisfy the condition. The middle integers are 36 and 106, and their difference is 70, not 0.
So if the video said the answer is 0, that&rsquo;s only true if the question asked for the minimum possible middle integer ,  but if it asks for the difference between highest and lowest possible, then the correct answer is 70, not 0.