When you take the product of any three consecutive positive integers, it is a universal truth that this product will always be divisible by 6. Here's why:
Let's label the three consecutive integers as n, n+1, and n+2 (where n is a positive integer). Then, the product of these three integers is expressed as:
n * (n+1) * (n+2)
Among these three integers:
- At least one of the integers must be even. In consecutive integers, you are guaranteed to hit an even number because they alternate between even and odd: if n is even, then n is divisible by 2; if n is odd, then n+1 is even, and therefore divisible by 2.
- At least one of the integers will be divisible by 3. This is because every third number is a multiple of 3, so when you have a sequence of three consecutive numbers, one of them must indeed be a multiple of 3.
Therefore, since one of the numbers is divisible by 2, and another is divisible by 3, the product of the three numbers will have factors of both 2 and 3, making the product divisible by 2*3 = 6.
Thus, no matter which group of three consecutive positive integers you choose, their product will always be divisible by 6.
Out of the options given:
c) no common factor - This is incorrect; the common factors are at least 2 and 3.
a) 4 - This is incorrect; although one number may be even, there is no guarantee that the even number is also a multiple of 4.
b) 6 - This is the correct answer; as shown above, the product is always divisible by 6.
d) only 1 - This is incorrect; other than 1, which is a divisor of every integer, the product also has 6 as a common divisor.
Therefore, the correct choice is:
b) 6
