Winter'2015'Math'211'Final'Exam'' ' ' ' Name_______________________________'''

Instructions:'Show%ALL%work.%Simplify'wherever'possible.'Clearly'indicate'your'final'answer.'

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'Problem'Number'

Points'Possible' Score'

1' 15' '

2' 20' '

3' 20' '

4' 15' '

5' 20' '

6' 10' '

7' 20' '

8' 20' '

9' 20' '

Subtotal' 160' '

Extra'credit' 15' '

Total' 160' '

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1) Let'A'be'the'set'of'integers'from'"5'to'5.'Define'a'relation'R'on'A'where' a, b( ) 'is'in'R'if' a2 + ab + b ≥ 0 '''a) Is'this'relation'reflexive?'Why'or'why'not?'

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b) Is'this'relation'symmetric?'Why'or'why'not?'''''''''''''''''

c) Is'this'relation'transitive?'Why'or'why'not?''

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2) Determine'if'the'number'111,000,111,000,111,000,111,000,111,000,111,000,111,000'is'divisible'by'the'following,'using'our'divisibility'rules'(show'work).''a) Divisible'by'3?'

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b) Divisible'by'4?'''''''''

c) Divisible'by'11?''''''''

d) Divisible'by'45?''''''''

e) The'number'111,000,111,000,111,000,111,000,111,000,111,000,111,00A'is'divisible'by'18.'What'number'must'A'be?'

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3) 'a) I'am'thinking'of'a'number.'The'least'common'multiple'between'my'number'and'108'is'1512.'The'

Greatest'Common'Factor'between'my'number'and'108'is'12.'What'number'am'I'thinking'of?''''''''''''''''''''''

b) A'rectangular'field'measures'160'yards'by'172'yards.''I'want'to'divide'the'field'into'squares,'each'the'same'size?''If'the'sides'of'these'squares,'measured'in'yards,'are'to'have'whole'number'lengths,'what'is'the'largest'possible'size'for'those'squares?'How'many'squares'of'this'size'will'there'be?''

'Hint:'Make'sure'to'answer'both'questions.'

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4) 'a) How'many'numbers'evenly'divide'672?'''''''''''''''''b) What'are'the'four'largest'such'numbers?'Justify'your'logic.'

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c) How'many'divisors'of'672'are'greater'than'67?'Justify'your'logic.'''''''''

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5) Determine'the'following,'using'the'lattice'method'for'part'a,'and'the'standard'algorithm'for'part'b.'''a) 1,121,101three ⋅102three '

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b) 10,53Aeleven ⋅35eleven '''''''''' '

6) Determine'the'following'using'the'base'2'version'of'the'“nine’s'compliment”'method.'Then,'check'your'answer'using'the'standard'algorithm.''a) 10,001,101two −1,001,110two '

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7) Determine'the'following.'In'each'case'the'remainder'should'be'zero.'Use'the'standard'algorithm'for'part'a,'and'the'scaffold'method'for'part'b.''a) 122,200,012three ÷101three '

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b) 1,013,214 five ÷ 321 five '' '

8) Evaluate'each'of'the'following'expressions.'To'justify'your'answer,'model'the'expression'using'positive'and'negative'chips.'Make'sure'to'model'the'given'expressions,'not'equivalent'expressions'that'use'different'operations.'To'represent'“putting'sets'together”'circle'each'individual'set.'To'represent'“taking'away”'circle'any'chips'that'you'are'removing.'(see'example'below)'

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a) 2 − 5 ''''''''

b) 2 + −5( ) '''''''''''

c) −2 × −5( ) '''''''''''

d) 2 × −5( ) '''''''''''

e) −2 × 5 '' '

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9) When'using'chips'to'model'division'where'at'least'one'of'the'numbers'in'the'problem'is'negative,'the'question'being'asked'determines'the'method'that'you'use.'Create,'answer'and'model'division'problems'(each'with'at'least'one'negative'number)'for'each'of'the'methods'below.'Make'sure'to'include'how'a'student'could'determine'the'answer'modeling'the'problem'in'this'way.''a) Measurement'Method'

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b) Partative'Method'''''''''''''''''

c) One'where'neither'the'measurement'nor'partative'method'is'appropriate,'so'instead'you'use'the'missing'factor'approach.'

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Extra Credit: The pyramid below is called a “prime pyramid.” Each row in the pyramid begins with a 1, and the nth row ends with the number n. In each row, the consecutive numbers from 1 to n are rearranged in such a way that the sum of any two adjacent numbers is prime. Complete the rest of the pyramid. 1

1 2 1 2 3 1 2 3 4 1 4 3 2 5 1 4 3 2 5 6

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