Module 2 Homework Problems
Complete the Following Problems
1.) True or False: \(5\vert 58123\)
2.) True or False: \(6\vert 361239\)
3.) True or False: \(7\vert 9173\)
4.) Show that \(13\vert 585\) using the definition of divides.
5.) Show that \(17\vert 2091\) using the definition of divides.
6.) Let \(n=312\) and \(m=5\). Find \(q\) and \(r\) such that \(n=mq+r\) and \(0\leq r<m\).
7.) Let \(n=522\) and \(m=3\). Find \(q\) and \(r\) such that \(n=mq+r\) and \(0\leq r<m\).
8.) Let \(n=7\) and \(m=15\). Find \(q\) and \(r\) such that \(n=mq+r\) and \(0\leq r<m\). Be careful! There might not be anything wrong with the answer you want to write!
9.) Compute \(23\ mod\ 5\).
10.) Compute \(541\ mod\ 6\).
11.) Compute \((31+81)\ mod\ 11\)
12.) Compute \(31-81\ mod\ 11\).
13.) Compute \(39\cdot 351\ mod 10\)
14.) Compute \((13)^{21}\ mod\ 12\)
15.) Compute \((13)^{20}\ mod\ 11\). (Hint: \(2^{20}=(2^5)^{4}\) Be patient with this one!)
16.) Compute \((2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7)\ mod\ 13\) (Hint: As the product of numbers is currently stated, the multiplication rule is useless because all factors are less than \(13\). Multiply pairs of numbers so that you get products greater than \(13\). Then try the multiplication rule.)
17.) Find the \(1\)’s digit of \((23)^{50}\). The hint from problem \(15\) helps.
18.) True or False: \(621\equiv 421\ (mod\ 3)\). Explain.
19.) True or False: \(720\equiv 450\ (mod\ 6)\). Explain.
20.) True or False: \(5123\equiv 17\ (mod\ 23)\). Explain.
21.) Find the set of all integers \(x\) such that \(x\equiv 1\ mod\ 5\). (Hint: if unsure how to do this, try to determine what the notation means in English. That almost always helps you get started on any problem).
22.) Is \(431\) prime? Support your answer.
23.) Is \(919\) prime? Support your answer.
24.) Factor \(920\) into a product of prime powers.
25.) Factor \(4012\) into a product of prime powers.
26.) What is the biggest prime number?
27.) Lets suppose that \(2,3,5\) and \(7\) were the only prime numbers in existence. Let \(Q_4=2\cdot3\cdot5\cdot7+1\). Use this number \(Q_4\) to argue why there must be more primes than \(2,3,5\) and \(7\). Try to do this without factoring anything and without using what you are trying to prove (e.g. don’t say anything like “well, 11 is prime too.” We are trying to use pure logic to show why there are more primes as opposed to simply listing another prime number.)
28.) Find \(gcd(45,12)\).
29.) Find \(lcm(34, 120)\).
30.) Find \(gcd(1205, 3505)\).
31.) Find \(lcm(450, 312)\).
32.) Let \(p\) and \(q\) be any two prime numbers. Find \(gcd(p,q)\) and \(lcm(p,q)\), explaining your answer in as much detail as you can.