Here are 3 problems that i couldn't solve them. I would wonder if anybody help me.
a) If prove that
b) If prove that
c) If are natural numbers and and , prove that
a) is false. For example, if a=2 and b=1 then 3a+4b = 10 (multiple of 5), but 2a^2-3ab+2b^2 = 4 (not a multiple of 25).
b) Suppose that p is one of the prime factors of a. Then p must also be a divisor of b. If is the highest power of p that divides a, and is the highest power of p that divides b, then the condition implies that , or . Conversely, if that condition holds for each prime divisor of a, then . The result then follows from the fact that .
c) Hint: notice that .