Question

1. Let a and b be integers. Prove that if a|b, then a”|6" for all positive integers n.

Answer 1

Solution

- a and b are integers. a | b means that integer a can divide integer b with no remainders.

- Let the Quotient of the division be k, so we can say:

$\begin{gathered} a|b=k \\ \\ \text{ Put in an easier way, we have:} \\ \frac{b}{a}=k \\ \\ where, \\ k\text{ is an integer since }a\text{ directly divides b} \end{gathered}$

- Now, we are asked to find

$a^n|b^n$

- Again, we can rewrite this as:

$\frac{b^n}{a^n}$

- We can rewrite this expression using the law of exponents that says:

$\frac{x^m}{y^m}=(\frac{x}{y})^m$

- Applying this law, we have:

$\frac{b^n}{a^n}=(\frac{b}{a})^n$

- But we already know that

$\frac{b}{a}=k$

- Thus, we have that:

$\begin{gathered} \frac{b^n}{a^n}=k^n \\ \\ That\text{ is,} \\ a^n|b^n=k^n \\ for\text{ all positive integers n} \\ \\ k^n\text{ is an integer as well because }k\text{ is an integer.} \end{gathered}$

- Therefore, we have successfully proved the assertion