Prove the following: 𝐹𝑜𝑟 𝑚 𝑎𝑛𝑑 𝑛 ∈ ℤ, 𝑖𝑓 𝑚 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 4 𝑎𝑛𝑑 𝑛 𝑖𝑠 𝑎𝑛𝑦 𝑒𝑣𝑒𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟,𝑡ℎ𝑒𝑛 𝑚 ∙ 𝑛 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 8.

Answer 1

Assume m is divisible by 4 and n is an even integer

Then, n is divisible by 2 by definition of even

Therefore, there exists some a, b integers such that 4a=m and 2b=n

Then, m*n=4a*2b

=8ab which is divisible by 8