Let a and b be any integer, if a|b then a^2|b^2. True or false?
If 2|4 then does 4|16 yes.. answers are 2 and 4
If 15|30 then does 225|900 answers are 2 and 4
If you think of 15 as 3 times 5 then squared gives you 3355
if you think of 30 as 2 times 3 times 5 then squared gives you 223355
Everything cancels except the 2 times 2 which is 4.
True
if a|b then b = ka for some integer k
b^2 = (ka)^2 = k^2 a^2 which is a multiple of a^2
To determine if the statement "If a|b, then a^2|b^2" is true or false, we need to analyze the meaning of the vertical bar symbol "|," which represents divisibility.
The statement "a|b" means that "a" divides "b" evenly, i.e., there exists an integer "k" such that "b = a * k." In other words, "b" is a multiple of "a."
Now let's consider the statement "a^2|b^2," which means that "a^2" divides "b^2" evenly. If this statement is true, it implies that there exists an integer "m" such that "b^2 = a^2 * m."
To determine whether the original statement is true or false, we need to prove it or find a counterexample.
Proof:
If a|b, then there exists an integer "k" such that "b = a * k." We can square both sides of this equation to get "b^2 = (a * k)^2 = a^2 * k^2." Since "k^2" is an integer, let's denote it as "m" for simplicity. Our equation becomes "b^2 = a^2 * m," which means "a^2|b^2."
Hence, the statement "If a|b, then a^2|b^2" is true.
In conclusion, if "a" divides "b" evenly, then "a^2" will also divide "b^2" evenly.