{"id":463,"date":"2017-10-28T22:23:36","date_gmt":"2017-10-28T22:23:36","guid":{"rendered":"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/?post_type=chapter&p=463"},"modified":"2017-12-24T03:31:26","modified_gmt":"2017-12-24T03:31:26","slug":"problem-bank-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/chapter\/problem-bank-2\/","title":{"raw":"Problem Bank","rendered":"Problem Bank"},"content":{"raw":"[latexpage]\r\n
\r\n

Problem 28<\/h3>\r\n
    \r\n \t
  1. If you were counting in base four, what number would you say just before you said $100_\\text{four}$?<\/li>\r\n \t
  2. What number is one more than $133_\\text{four}$?<\/li>\r\n \t
  3. What is the greatest three-digit number that can be written in base four? What numbers come just before and just after that number?<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
    \r\n

    Problem 29<\/h3>\r\n

    Explain what is wrong with writing\u00a0$313_\\text{two}$\u00a0or $28_\\text{eight}$.<\/p>\r\n\r\n<\/div>\r\n \r\n

    \r\n

    Problem 30<\/h3>\r\n
      \r\n \t
    1. Write out the base three numbers from\u00a0$1_\\text{three}$\u00a0to $200_\\text{three}$.<\/li>\r\n \t
    2. Write out the base five numbers from\u00a0$1_\\text{five}$\u00a0to $100_\\text{five}$.<\/li>\r\n \t
    3. Write the four base six numbers that come after\u00a0$154_\\text{six}$.<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
      \r\n

      Problem 31<\/h3>\r\n

      Convert each base ten number to a base four\u00a0number. Explain how you did it.<\/p>\r\n\\[ 13, \\qquad\\qquad 8,\\qquad\\qquad 24,\\qquad\\qquad 49 \\]\r\n

      Challenges:<\/b><\/p>\r\n\\[ 0.125, \\qquad\\qquad 0.11111\\dots = 0.\\overline{1} \\]\r\n\r\n<\/div>\r\n \r\n

      \r\n

      Problem 32<\/h3>\r\n

      In order to use base sixteen, we need sixteen digits \u2014 they will represent the numbers zero through fifteen.\u00a0 We can use our usual digits 0\u20139, but we need new symbols<\/i> to represent the digits<\/i> ten, eleven, twelve, thirteen, fourteen, and fifteen.\u00a0 Here's one standard convention:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      base ten<\/strong><\/th>\r\nbase sixteen<\/strong><\/th>\r\n<\/tr>\r\n
      7<\/td>\r\n$7_\\text{sixteen}$<\/td>\r\n<\/tr>\r\n
      8<\/td>\r\n$8_\\text{sixteen}$<\/td>\r\n<\/tr>\r\n
      9<\/td>\r\n$9_\\text{sixteen}$<\/td>\r\n<\/tr>\r\n
      10<\/td>\r\n$\\text{A}_\\text{sixteen}$<\/td>\r\n<\/tr>\r\n
      11<\/td>\r\n$\\text{B}_\\text{sixteen}$<\/td>\r\n<\/tr>\r\n
      12<\/td>\r\n$\\text{C}_\\text{sixteen}$<\/td>\r\n<\/tr>\r\n
      13<\/td>\r\n$\\text{D}_\\text{sixteen}$<\/td>\r\n<\/tr>\r\n
      14<\/td>\r\n$\\text{E}_\\text{sixteen}$<\/td>\r\n<\/tr>\r\n
      15<\/td>\r\n$\\text{F}_\\text{sixteen}$<\/td>\r\n<\/tr>\r\n
      16<\/td>\r\n$10_\\text{sixteen}$<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
        \r\n \t
      1. Convert these numbers from base sixteen to base ten, and show your work:\r\n\\[ 6\\textrm{D}_{\\text{sixteen}}\r\n\\qquad \\quad\r\n\\textrm{AE}_{\\text{sixteen}}\r\n\\qquad\\quad\r\n9\\textrm{C}_{\\text{sixteen}}\r\n\\qquad\\quad\r\n2\\textrm{B}_{\\text{sixteen}} \\]<\/li>\r\n \t
      2. Convert these numbers from base ten to base sixteen, and show your work:\r\n\\[97\r\n\\qquad \\quad\r\n144\r\n\\qquad\\quad\r\n203\r\n\\qquad\\quad\r\n890 \\]<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
        \r\n

        Problem 33<\/h3>\r\n

        How many different symbols would you need for a base twenty-five system? Justify your answer.<\/p>\r\n\r\n<\/div>\r\n \r\n

        \r\n

        Problem 34<\/h3>\r\n

        All of the following numbers are multiples of three.<\/p>\r\n\\[3, \\quad 6, \\quad 9, \\quad 12, \\quad 21, \\quad 27, \\quad 33, \\quad 60, \\quad 81, \\quad 99.\\]\r\n

          \r\n \t
        1. Identify the powers of <\/i>3\u00a0in the list. Justify your answer.<\/li>\r\n \t
        2. Write each of the numbers above in base three.<\/li>\r\n \t
        3. In base three: how can you recognize a multiple of <\/i>3?\u00a0 Explain your answer.<\/li>\r\n \t
        4. In base three: how can you recognize a power of <\/i>3? Explain your answer.<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
          \r\n

          Problem 35<\/h3>\r\n

          All of the following numbers are multiples of five.<\/p>\r\n\\[5, \\quad 10, \\quad 15, \\quad 25, \\quad 55, \\quad 75, \\quad 100, \\quad 125, \\quad 625, \\quad 1000.\\]\r\n

            \r\n \t
          1. Identify the powers of <\/i>5 in the list. Justify your answer.<\/li>\r\n \t
          2. Write each of the numbers above in base five.<\/li>\r\n \t
          3. In base five: how can you recognize a multiple of <\/i>5?\u00a0 Explain your answer.<\/li>\r\n \t
          4. In base five: how can you recognize a power of <\/i>5? Explain your answer.<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
            \r\n

            Problem 36<\/h3>\r\n

            Convert each number to the given base.<\/p>\r\n\r\n

              \r\n \t
            1. $395_\\text{ten}$ into base eight.<\/li>\r\n \t
            2. $52_\\text{ten}$ into base two.<\/li>\r\n \t
            3. $743_\\text{ten}$ into base five.<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
              \r\n

              Problem 37<\/h3>\r\n

              What bases makes theses equations true? Justify your answers.<\/p>\r\n\r\n

                \r\n \t
              1. $35 = 120_{\\underline{\\quad}}$<\/li>\r\n \t
              2. $41_{\\text{six}} = 27_{\\underline{\\quad}}$<\/li>\r\n \t
              3. $ 52_{\\text{seven}} = 34_{\\underline{\\quad}}$<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
                \r\n

                Problem 38<\/h3>\r\n

                What bases makes theses equations true? Justify your answers.<\/p>\r\n\r\n

                  \r\n \t
                1. $32 = 44_{\\underline{\\quad}} $<\/li>\r\n \t
                2. $57_{\\text{eight}} = 10_{\\underline{\\quad}}$<\/li>\r\n \t
                3. $ 31_{\\text{four}} = 11_{\\underline{\\quad}}$<\/li>\r\n \t
                4. $15_x = 30_y\u00a0 $<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
                  \r\n

                  Problem 39<\/h3>\r\n
                    \r\n \t
                  1. Find a base ten number that is twice the product of its two digits. Is there more than one answer? Justify what you say.<\/li>\r\n \t
                  2. Can you solve this problem in any base other than ten?<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
                    \r\n

                    Problem 40<\/h3>\r\n
                      \r\n \t
                    1. I have a four-digit number written in base ten. When I multiply my number by four, the digits get reversed.\u00a0 Find the number.<\/li>\r\n \t
                    2. Can you solve this problem in any base other than ten?<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
                      \r\n

                      Problem 41<\/h3>\r\n

                      Convert each base four number to a base ten number. Explain how you did it.<\/p>\r\n\\[ 13_{\\text{four}}\r\n\\qquad\r\n322_{\\text{four}}\r\n\\qquad\r\n101_{\\text{four}}\r\n\\qquad\r\n1300_{\\text{four}}\\]\r\n

                      Challenges:<\/b><\/p>\r\n\\[ 0.2_{\\text{four}}\r\n\\qquad\\qquad\r\n0.111..._{\\text{four}}=0.\\overline{1}_{\\text{four}}\\]\r\n\r\n<\/div>\r\n \r\n

                      \r\n

                      Problem 42<\/h3>\r\n

                      Consider this base ten number (I got this by writing the numbers from 1 to 60 in order next to one another):<\/p>\r\n\\[12345678910111213\\ldots57585960.\\]\r\n

                        \r\n \t
                      1. What is the largest number that can be produced by erasing one hundred digits of the number? (When you erase a digit it goes away. For example, if you start with the number\u00a012345\u00a0and erase the middle digit, you produce the number 1245.) How do you know<\/i> you got the largest possible number?<\/li>\r\n \t
                      2. What is the smallest number that can be produced by erasing one hundred digits of the number? How do you know<\/i> you got the smallest possible number?<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
                        \r\n

                        Problem 43<\/h3>\r\n

                        Can you find two different numbers (not necessarily single digits!)\u00a0$a$\u00a0and\u00a0$b$\u00a0so that\u00a0$a_b = b_a$? Can you find more than one solution? Justify your answers.<\/p>\r\n\r\n<\/div>","rendered":"

                        \n

                        Problem 28<\/h3>\n
                          \n
                        1. If you were counting in base four, what number would you say just before you said \"100_\text{four}\"?<\/li>\n
                        2. What number is one more than \"133_\text{four}\"?<\/li>\n
                        3. What is the greatest three-digit number that can be written in base four? What numbers come just before and just after that number?<\/li>\n<\/ol>\n<\/div>\n

                           <\/p>\n

                          \n

                          Problem 29<\/h3>\n

                          Explain what is wrong with writing\u00a0\"313_\text{two}\"\u00a0or \"28_\text{eight}\".<\/p>\n<\/div>\n

                           <\/p>\n

                          \n

                          Problem 30<\/h3>\n
                            \n
                          1. Write out the base three numbers from\u00a0\"1_\text{three}\"\u00a0to \"200_\text{three}\".<\/li>\n
                          2. Write out the base five numbers from\u00a0\"1_\text{five}\"\u00a0to \"100_\text{five}\".<\/li>\n
                          3. Write the four base six numbers that come after\u00a0\"154_\text{six}\".<\/li>\n<\/ol>\n<\/div>\n

                             <\/p>\n

                            \n

                            Problem 31<\/h3>\n

                            Convert each base ten number to a base four\u00a0number. Explain how you did it.<\/p>\n

                              <\/span>   <\/span>\"\[ 13, \qquad\qquad 8,\qquad\qquad 24,\qquad\qquad 49 \]\"<\/p>\n

                            Challenges:<\/b><\/p>\n

                              <\/span>   <\/span>\"\[ 0.125, \qquad\qquad 0.11111\dots = 0.\overline{1} \]\"<\/p>\n<\/div>\n

                             <\/p>\n

                            \n

                            Problem 32<\/h3>\n

                            In order to use base sixteen, we need sixteen digits \u2014 they will represent the numbers zero through fifteen.\u00a0 We can use our usual digits 0\u20139, but we need new symbols<\/i> to represent the digits<\/i> ten, eleven, twelve, thirteen, fourteen, and fifteen.\u00a0 Here’s one standard convention:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                            base ten<\/strong><\/th>\nbase sixteen<\/strong><\/th>\n<\/tr>\n
                            7<\/td>\n\"7_\text{sixteen}\"<\/td>\n<\/tr>\n
                            8<\/td>\n\"8_\text{sixteen}\"<\/td>\n<\/tr>\n
                            9<\/td>\n\"9_\text{sixteen}\"<\/td>\n<\/tr>\n
                            10<\/td>\n\"\text{A}_\text{sixteen}\"<\/td>\n<\/tr>\n
                            11<\/td>\n\"\text{B}_\text{sixteen}\"<\/td>\n<\/tr>\n
                            12<\/td>\n\"\text{C}_\text{sixteen}\"<\/td>\n<\/tr>\n
                            13<\/td>\n\"\text{D}_\text{sixteen}\"<\/td>\n<\/tr>\n
                            14<\/td>\n\"\text{E}_\text{sixteen}\"<\/td>\n<\/tr>\n
                            15<\/td>\n\"\text{F}_\text{sixteen}\"<\/td>\n<\/tr>\n
                            16<\/td>\n\"10_\text{sixteen}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                              \n
                            1. Convert these numbers from base sixteen to base ten, and show your work:\n

                                <\/span>   <\/span>\"\[ 6\textrm{D}_{\text{sixteen}}<\/p>\n<\/li>\n

                            2. Convert these numbers from base ten to base sixteen, and show your work:\n

                                <\/span>   <\/span>\"\[97<\/p>\n<\/li>\n<\/ol>\n<\/div>\n

                               <\/p>\n

                              \n

                              Problem 33<\/h3>\n

                              How many different symbols would you need for a base twenty-five system? Justify your answer.<\/p>\n<\/div>\n

                               <\/p>\n

                              \n

                              Problem 34<\/h3>\n

                              All of the following numbers are multiples of three.<\/p>\n

                                <\/span>   <\/span>\"\[3, \quad 6, \quad 9, \quad 12, \quad 21, \quad 27, \quad 33, \quad 60, \quad 81, \quad 99.\]\"<\/p>\n

                                \n
                              1. Identify the powers of <\/i>3\u00a0in the list. Justify your answer.<\/li>\n
                              2. Write each of the numbers above in base three.<\/li>\n
                              3. In base three: how can you recognize a multiple of <\/i>3?\u00a0 Explain your answer.<\/li>\n
                              4. In base three: how can you recognize a power of <\/i>3? Explain your answer.<\/li>\n<\/ol>\n<\/div>\n

                                 <\/p>\n

                                \n

                                Problem 35<\/h3>\n

                                All of the following numbers are multiples of five.<\/p>\n

                                  <\/span>   <\/span>\"\[5, \quad 10, \quad 15, \quad 25, \quad 55, \quad 75, \quad 100, \quad 125, \quad 625, \quad 1000.\]\"<\/p>\n

                                  \n
                                1. Identify the powers of <\/i>5 in the list. Justify your answer.<\/li>\n
                                2. Write each of the numbers above in base five.<\/li>\n
                                3. In base five: how can you recognize a multiple of <\/i>5?\u00a0 Explain your answer.<\/li>\n
                                4. In base five: how can you recognize a power of <\/i>5? Explain your answer.<\/li>\n<\/ol>\n<\/div>\n

                                   <\/p>\n

                                  \n

                                  Problem 36<\/h3>\n

                                  Convert each number to the given base.<\/p>\n

                                    \n
                                  1. \"395_\text{ten}\" into base eight.<\/li>\n
                                  2. \"52_\text{ten}\" into base two.<\/li>\n
                                  3. \"743_\text{ten}\" into base five.<\/li>\n<\/ol>\n<\/div>\n

                                     <\/p>\n

                                    \n

                                    Problem 37<\/h3>\n

                                    What bases makes theses equations true? Justify your answers.<\/p>\n

                                      \n
                                    1. \"35 = 120_{\underline{\quad}}\"<\/li>\n
                                    2. \"41_{\text{six}} = 27_{\underline{\quad}}\"<\/li>\n
                                    3. \"52_{\text{seven}} = 34_{\underline{\quad}}\"<\/li>\n<\/ol>\n<\/div>\n

                                       <\/p>\n

                                      \n

                                      Problem 38<\/h3>\n

                                      What bases makes theses equations true? Justify your answers.<\/p>\n

                                        \n
                                      1. \"32 = 44_{\underline{\quad}}\"<\/li>\n
                                      2. \"57_{\text{eight}} = 10_{\underline{\quad}}\"<\/li>\n
                                      3. \"31_{\text{four}} = 11_{\underline{\quad}}\"<\/li>\n
                                      4. \"15_x = 30_y \"<\/li>\n<\/ol>\n<\/div>\n

                                         <\/p>\n

                                        \n

                                        Problem 39<\/h3>\n
                                          \n
                                        1. Find a base ten number that is twice the product of its two digits. Is there more than one answer? Justify what you say.<\/li>\n
                                        2. Can you solve this problem in any base other than ten?<\/li>\n<\/ol>\n<\/div>\n

                                           <\/p>\n

                                          \n

                                          Problem 40<\/h3>\n
                                            \n
                                          1. I have a four-digit number written in base ten. When I multiply my number by four, the digits get reversed.\u00a0 Find the number.<\/li>\n
                                          2. Can you solve this problem in any base other than ten?<\/li>\n<\/ol>\n<\/div>\n

                                             <\/p>\n

                                            \n

                                            Problem 41<\/h3>\n

                                            Convert each base four number to a base ten number. Explain how you did it.<\/p>\n

                                              <\/span>   <\/span>\"\[ 13_{\text{four}}<\/p>\n

                                            Challenges:<\/b><\/p>\n

                                              <\/span>   <\/span>\"\[ 0.2_{\text{four}}<\/p>\n<\/div>\n

                                             <\/p>\n

                                            \n

                                            Problem 42<\/h3>\n

                                            Consider this base ten number (I got this by writing the numbers from 1 to 60 in order next to one another):<\/p>\n

                                              <\/span>   <\/span>\"\[12345678910111213\ldots57585960.\]\"<\/p>\n

                                              \n
                                            1. What is the largest number that can be produced by erasing one hundred digits of the number? (When you erase a digit it goes away. For example, if you start with the number\u00a012345\u00a0and erase the middle digit, you produce the number 1245.) How do you know<\/i> you got the largest possible number?<\/li>\n
                                            2. What is the smallest number that can be produced by erasing one hundred digits of the number? How do you know<\/i> you got the smallest possible number?<\/li>\n<\/ol>\n<\/div>\n

                                               <\/p>\n

                                              \n

                                              Problem 43<\/h3>\n

                                              Can you find two different numbers (not necessarily single digits!)\u00a0\"a\"\u00a0and\u00a0\"b\"\u00a0so that\u00a0\"a_b = b_a\"? Can you find more than one solution? Justify your answers.<\/p>\n<\/div>\n","protected":false},"author":21,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-463","chapter","type-chapter","status-publish","hentry"],"part":134,"_links":{"self":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/463","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":22,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/463\/revisions"}],"predecessor-version":[{"id":1796,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/463\/revisions\/1796"}],"part":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/parts\/134"}],"metadata":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/463\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/media?parent=463"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapter-type?post=463"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/contributor?post=463"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/license?post=463"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}