1. ## Divisibly Problems

Here are 3 problems that i couldn't solve them. I would wonder if anybody help me.

a) If $\displaystyle 5|3a+4b$ prove that $\displaystyle 25|2a^2-3ab+2b^2$
b) If $\displaystyle a^9|b^5$ prove that $\displaystyle a^7|b^4$
c) If $\displaystyle a, b, c$ are natural numbers and $\displaystyle a^b|b^a$ and $\displaystyle b^c|c^b$, prove that $\displaystyle a^c|c^a$

2. Originally Posted by havaliza
Here are 3 problems that i couldn't solve them. I would wonder if anybody help me.

a) If $\displaystyle 5|3a+4b$ prove that $\displaystyle 25|2a^2-3ab+2b^2$
b) If $\displaystyle a^9|b^5$ prove that $\displaystyle a^7|b^4$
c) If $\displaystyle a, b, c$ are natural numbers and $\displaystyle a^b|b^a$ and $\displaystyle b^c|c^b$, prove that $\displaystyle a^c|c^a$
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 $\displaystyle p^s$ is the highest power of p that divides a, and $\displaystyle p^t$ is the highest power of p that divides b, then the condition $\displaystyle a^9|b^5$ implies that $\displaystyle 9s\leqslant5t$, or $\displaystyle \tfrac st\leqslant\tfrac59$. Conversely, if that condition holds for each prime divisor of a, then $\displaystyle a^9|b^5$. The result then follows from the fact that $\displaystyle \tfrac59<\tfrac47$.

c) Hint: notice that $\displaystyle a^{bc} | b^{ac} | c^{ba}$.

3. Thanks for the help. The first problem was right but i had mistyped it

a) If $\displaystyle 5|3a+4b$ prove that $\displaystyle 25|2a^2-3ab-2b^2$

4. Originally Posted by havaliza
Thanks for the help. The first problem was right but i had mistyped it

a) If $\displaystyle 5|3a+4b$ prove that $\displaystyle 25|2a^2-3ab-2b^2$
In that case, notice that $\displaystyle 2a^2-3ab-2b^2 = (a+2b)(2a-b)$, and show that each of those factors can be expressed as a multiple of 3a+4b plus some multiple of 5.