Prove that the value of the expression is divisible by 63.

Question

Prove that the value of the expression: (16^3–8^3)(4^3+2^3) is divisible by 63.

Answer 1

To be divisible by 63, it must be divisible by both 7 and 9, since 7x9 = 63.  Factor the expression, simplify, and see if both 7 and 9 are factors, which would make it divisible by both 7 and 9.

 

(16³ - 8³)(4³ + 2³) = [(2⁴)³ - (2³)³] [(2²)³ + 2³]

 

= (2¹² - 2⁹)(2⁶ + 2³) = [2⁹(2³ - 1)] [2³(2³ + 1)]

 

= [2⁹(8 - 1)] [2³(8 + 1)] = 2⁹(7) 2³ (9) = 63(2¹²)

 

It is a multiple of 63, and hence divisible by 63.

Answer 2

• 16^3 - 8^3 is equivalent to 2^3 -1^3 modulo 7

• 2^3 - 1^3 = 7, so 16^3 - 8^3 is equivalent to 0 modulo 7.

Therefore, 7 divides (16^3 - 8^3). 

4^3 + 2^3 = 64 + 8 = 72, so 9 divides (4^3 + 2^3). 

Thus 63 divides (16^3 - 8^3)(4^3 + 2^3).