MATH211 Assignment 3 with Solution (2011)

Math211

Assignment3

Due13October2011

1.An apartment building contains apartments at two rental rates:Those of Type A [4]

rent for$650per month and those of Type B for$875per month.When all the

apartments are rented,the gross income is$24,800.How many apartments are there

of each type?(Give all possibilities and justify your answer.)

2.Find the general integer solution of each of the following Diophantine equations:

[3]

(a)4x+6y−9z=10;(b)3x+9y+12z=16.

3.Find all the non-negative solutions x≥0,y≥0and z≥0of the equation

[2]

2x+3y+5z=5.

(You do not have tofind the general integer solution,but you do have to justify that

you found all the non-negative solutions.)

4.When Mr.Smith returned from a trip in Europe in1966,he found that he had in [4]

his possession35British sixpence coins,55French ten-centimes pieces and77Greek

drachmas.Mr.Smith converted each of theses coins to its value in Canadian money

(rounded offto the nearest cent)and found that the total was worth$5.86.How

much was each coin worth in1966(to the nearest cent)?

e the sieve of Eratosthenes tofind all the prime numbers less than120,and list all [2]

the twin prime pairs in that range.

6.Find the prime factorization of the following integers by hand(show your work):

[2]

(a)1728;(b)684600.

7.Let a and b be non-zero integers and put g=gcd(a,b)and m=ab

g .Show that

[3]

(a)a|m and b|m.

(b)If k is any integer such that a|k and b|k,then m|k.[Hint:Use Euclid’s Lemma.]

Note:m=ab

gcd(a,b)

is called the least common multiple of a and b.