Answer
See explanations.
Work Step by Step
Step 1. Prove the statement is true for $n=1$: test for $r=0,1$
$\begin{pmatrix} 1\\0 \end{pmatrix}=\frac{1}{0!(1-0)!}=1$ which is an integer.
$\begin{pmatrix} 1\\1 \end{pmatrix}=\frac{1}{1!(1-1)!}=1$ which is an integer.
Step 2. Assume the statement is true for $n=k$:
$\begin{pmatrix} k\\r \end{pmatrix}$ is an integer for all $0\leq r\leq k$.
Step 3. Prove that it is true for $n=k+1$: (use the results from Exercise 53)
$\begin{pmatrix} k+1\\r \end{pmatrix}=\begin{pmatrix} k\\r \end{pmatrix}+\begin{pmatrix} k\\r-1 \end{pmatrix}=integer_1 + integer_2=integer_3$
Step 4. With mathematical induction, we proved that the statement is true for all n,
