the factors of 210 are 2,3,5,7

16*15*14 is is also consecutive numbers that is a multiple of 210

Shouldnt answer be 15-6=9?

Question

the factors of 210 are 2,3,5,7

16*15*14 is is also consecutive numbers that is a multiple of 210

Shouldnt answer be 15-6=9?

Simran Mehta · Accepted Answer

Solution:We need to find the difference between the highest and the lowest possible value of the middle integer in three consecutive positive integers whose product is a multiple of 210.
Factorize 210 210 = 2 * 3 * 5 * 7
Consider the product of three consecutive integers Let the three consecutive integers be n-1, n, and n+1, where n is the middle integer. The product of these three integers is: (n-1) * n * (n+1)
Determine the divisibility by 210 To be divisible by 210, the product must have factors 2, 3, 5, and 7.
Identify the values of n that satisfy the conditions For the product to be divisible by 2, 3, 5, and 7, at least one of n-1, n, or n+1 must be divisible by 2, 3, 5, and 7 respectively.
Start with n and check the multiples Let's try n = 7: (7-1) * 7 * (7+1) = 6 * 7 * 8 = 336 (not divisible by 5) Let's try n = 10: (10-1) * 10 * (10+1) = 9 * 10 * 11 = 990 (not divisible by 7) Let's try n = 14: (14-1) * 14 * (14+1) = 13 * 14 * 15 = 2730 (divisible by 2, 3, 5, and 7) So, n = 14 works.
Identify the highest possible value of n A quick check shows that for n = 34: (33 * 34 * 35) = 39270 (divisible by 2, 3, 5, and 7)
Calculate the difference The difference between the highest and lowest possible values of n: 34 - 14 = 20
Answer: The difference between the highest and lowest possible values of the middle integer is 20.