 ## MATH 2170 Number Theory 1 Assignment 1

### SHOW ALL WORK to get full marks. Leave answers as exact answers. For example, leave it as fractions such as 1{7 as opposed to decimals such as 0.142857. Word problems should have sentence answers with units. Fractions should be in lowest terms.  代写数论作业

As theory will be tested, quality of writing will be considering in marking. Only single display calculators are permitted.

All assignments must be handed in via Crowdmark by 11:59pm on  January  28th. Late assignments  will  not  be  accepted.  Failure  to follow the instructions will result in  a mark of 0.  代写数论作业

This assignment covers sections 1.1,1.2,1.3,2.1 and 2.2 in the textbook.

For questions 1–2, do not use prime factorization.

###  1. Show that if a| band c| d, then ac| bd.

 2. (For this question, do not use prime factorization)

Suppose that( e, f) and d are positive integers with d | ef. Prove that there exists positive integers a and d such that a | e,  b | f and d =ab. Further  show that the values of a and b are uniqueb if p(e, f)=1.

###  3. (a) What is the remainder when 112020 is divided by 17? A single line display calculator may be used, but nothing else. Show all your stepsd.  代写数论作业

(b) What is the last two  digits of 72020.  A single line display calculator may  be used, but nothing else. Show the work done.

 4. Prove that n12 a12 is divisible by 91 if n and a are both relatively prime to 91.

 5. Show that if p is an odd prime, then and