)lukaszh said:
The statement "7|(a^2+b^2)" means that 7 is a divisor of the sum of the squares of two numbers, a and b.
When 7 divides a number, it means that the number can be evenly divided by 7 without any remainder.
It is important to show that 7|a and 7|b because it proves that 7 is a common divisor of both a and b, which supports the original statement that 7 is a divisor of the sum of their squares.
Yes, this statement can be rewritten as "If 7 does not divide a or 7 does not divide b, then 7 does not divide (a^2+b^2)." This is the contrapositive form of the original statement.
The proof for this statement involves using the properties of divisibility and the fact that any number can be written as either 7k, 7k+1, 7k+2, 7k+3, 7k+4, 7k+5, or 7k+6 for some integer k. By substituting these values into the equation (a^2+b^2) and showing that they all result in a multiple of 7, the statement can be proven to be true.