- A number such as 11, 111, 1111 is called a repunit. Clearly eleven is a prime repunit. Find another one.
- Find the prime numbers less than 100 using the Sieve of Eratosthenes - make sure you actually draw it! Every math student should do this once - and only once.
- Prove that if \(a,b>0\) then \(\gcd(a,b)\text{lcm}(a,b)=ab\) using the FTA.
- Prove that if \(\gcd(a,b)=1\) and \(a|bc\) then \(a|c\) as well, using the FTA.
- Prove using the FTA that if \(gcd(a,b)=d\) then \(gcd\Big(\frac{a}{d},\frac{b}{d}\Big)=1\).
- How would you describe a factorization of a rational number? Do you think you could extend the FTA to this case? If so, how? If not, why would it not be appropriate?
- Show that if \(a\) and \(b\) are positive integers and \(a^3\mid b^2\), then \(a\mid b\).
- Is it possible for \(n!\) to end in exactly five zeros?
- Show that \(\log_{10} (5)\) is irrational.
- Show that \(3^{2/3}\) is irrational.
- By hand, find the prime factorizations of 36, 756, and 1001. Use these to find their pairwise gcds and lcms.
- By hand, find the gcd and lcm of \(2^2\cdot 3^5\cdot 7^2\cdot 13\cdot 37\) and \(2^3\cdot 3^4\cdot 11\cdot 31^2\).
- By any method you like, find the prime factorizations of \(2^{24}-1\) and \(10^8-1\), as well as their gcd.
- Find three prime repunits.
- Prove using the FTA that if \(gcd(a,b)=d\) then \(gcd\left(\frac{a}{d},\frac{b}{d}\right)=1\).
- Show that if \(a\) and \(b\) are positive integers such that \(a^3|b^2\), then \(a|b\).
- Show that if \(p^a\parallel m\) and \(p^b\parallel n\), then \(p^{a+b}\parallel mn\).
- Is it possible for \(n!\) to end in exactly five zeros?
- By hand, find the prime factorization of \(36\), \(756\), and \(1001\), and their pairwise gcd and lcm.
