intro to number theory, Exercises of Number Theory

Thm 1.3

(PF) Let a, b, and c ∈ ℤ and 𝑎|𝑏 𝑎𝑛𝑑 𝑎|𝑐. We need to show 𝑎|𝑏𝑐.

By definition of divisibility, ∃ 𝑘, 𝑙 ∈ ℤ 𝑠. 𝑡. 𝑏 = 𝑎𝑘 𝑎𝑛𝑑 𝑐 = 𝑎𝑙.

Now,

𝑏𝑐 = (𝑎𝑘)(𝑎𝑙) by substitution

= 𝑎(𝑎𝑘𝑙) by algebra

Note that 𝑎𝑘𝑙 ∈ ℤ, 𝑠𝑖𝑛𝑐𝑒 𝑎, 𝑘, 𝑙 ∈ ℤ 𝑎𝑛𝑑 ℤ 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑 𝑢𝑛𝑑𝑒𝑟 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛.

∴ 𝑎|𝑏𝑐 by definition of divisibility, which is what was to be proven. ∎