Answers:

1. Since n = 2x + 3,

n² - 1 = (2x + 3)² - 1 = 4x² + 12x + 3² - 1

 = 4x² + 12x + 8

 = 4 (x² + 3x + 2)     by factorizing 4


Therefore, n² - 1 is divisible by 4 where n = 2x + 3.

2. Is 6 a factor of 141x * 4y?

141x = 3 * 47x     by factorizing 3

4y = 2 * 2y           by factorizing 2

hence, 141x * 4y = (3 * 47x) * (2 * 2y)     by substitution

                  = 3 * 2 * 47x * 2y

                  = 6 * 47x * 2y


Therefore, 6 is a factor of 141x * 4y.

3. As x divides y and y divides z, by the definition of divisibility,

y = x * a and z = y * b for some integers a and b.

z = y * b

   = (x * a) * b     by substitution

   = x * (a * b) where a * b is an integer (the product of two integers is an integer).

Let p = a * b, z = x * p where p is an integer. By the definition of divisibility, x divides z.