Question

If p is a positive integer, then p(p + 1)(p − 1) is always divisible by

F. 7
G. 5
H. 4
J. 3
K. None of these

Answer 1

K

Explanation:

We can factor the given expression as:

p(p + 1)(p - 1) = p^3 - p

Notice that p^3 - p is the difference of two consecutive cubes. This can be factored further using the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2):

p^3 - p = p(p^2 - 1) = p(p + 1)(p - 1)

So, we have shown that p(p + 1)(p - 1) is equal to p^3 - p, which is the product of three consecutive integers. By definition, this product is always divisible by 3.

However, we cannot conclude that p(p + 1)(p - 1) is divisible by 4, 5, or 7 for all positive integers p. Therefore, the answer is K) None of these.