P(x)= the number of distinct prime factors of x

An integer 14ⁿ is divisible by 32 with remainder zero.

If x #y = (x + y)(x - y) for all real numbers, then which of the following must be true?

I. x #y = y #x

II. x #0 = 0 #x = x^2

III. x #-y = x #y

If the operation @ is defined for all numbers x and y by the equation x@y=, then 3@(11@2)=

Let x⊗y be defined as the product of the integers from x to y. For example, 3⊗7=3∗4∗5∗6∗7. What is the value (2⊗5)/(6⊗8)?

For all integers a and b, where a≠b, a★b=∣∣

What is the value of 4★2 ?

For each positive integer n, p(n) is defined to be the product of the digits of n. For example, p(724)=56 since 7∗2∗4=56.

Which of the following statements must be true?

I. p(10n)=p(n)

II. p(n+1)>p(n)

III. p(2n)=2p(n)

The operation ⊗ is defined for all nonzero numbers a and b by a ⊗ b = a/b – b/a. If x and y are nonzero numbers, which of the following statements must be true?

I. x ⊗ xy = x(1 ⊗ y)

II. x ⊗ y = -(y ⊗ x)

III. 1/x ⊗ 1/y = y ⊗ x

If A◊B=4A−B, what is the value of (3◊2)◊3?

For which of the following functions f is f(x) = f(1-x) for all x?