{"id":24528,"date":"2025-06-18T15:57:55","date_gmt":"2025-06-18T15:57:55","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=24528"},"modified":"2025-06-18T15:58:13","modified_gmt":"2025-06-18T15:58:13","slug":"the-decimal-equivalent-of-17-27-is-a-repeating-decimal","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/the-decimal-equivalent-of-17-27-is-a-repeating-decimal\/","title":{"rendered":"The decimal equivalent of (17)\/(27) is a repeating decimal"},"content":{"rendered":"\n

The decimal equivalent of (17)\/(27) is a repeating decimal. How many digits are in the repeating part of the decimal? (A) 1 (C) 3 (B) 2 (D) 6<\/p>\n\n\n\n

The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n

The correct answer is (D) 6<\/strong>.<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

To determine the length of the repeating decimal part for a fraction ab\\frac{a}{b} (in simplest form), focus on the denominator bb, specifically the part that is coprime with 10 (i.e., ignoring factors of 2 and 5 since they only affect terminating decimals).<\/p>\n\n\n\n

Here, the fraction is 1727\\frac{17}{27}.<\/p>\n\n\n\n

    \n
  1. Factor the denominator:<\/strong>
    27=3327 = 3^3<\/li>\n\n\n\n
  2. Since 27 is only composed of 3’s and has no factors of 2 or 5, the decimal will be repeating.<\/strong><\/li>\n\n\n\n
  3. Length of repeating decimal (period):<\/strong>
    The length of the repeating cycle (period) is the smallest positive integer kk such that: 10k\u22611(mod27)10^k \\equiv 1 \\pmod{27}<\/li>\n\n\n\n
  4. Calculate powers of 10 modulo 27:<\/strong>\n
      \n
    • 101mod\u2009\u200927=1010^1 \\mod 27 = 10 (not 1)<\/li>\n\n\n\n
    • 102mod\u2009\u200927=100mod\u2009\u200927=100\u22123\u00d727=100\u221281=1910^2 \\mod 27 = 100 \\mod 27 = 100 – 3 \\times 27 = 100 – 81 = 19 (not 1)<\/li>\n\n\n\n
    • 103mod\u2009\u200927=102\u00d710=19\u00d710=190mod\u2009\u20092710^3 \\mod 27 = 10^2 \\times 10 = 19 \\times 10 = 190 \\mod 27
      190\u22127\u00d727=190\u2212189=1190 – 7 \\times 27 = 190 – 189 = 1 \u2713<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n

      This means k=3k = 3 would give 103\u22611(mod27)10^3 \\equiv 1 \\pmod{27}.<\/p>\n\n\n\n

      However, this suggests a period of 3 digits, but we have to check the order carefully because of the power of 3:<\/p>\n\n\n\n

      Actually, the order for 3n3^n with n\u22652n \\geq 2 can be larger.<\/p>\n\n\n\n

      Checking further:<\/p>\n\n\n\n

        \n
      • For 27, the length of the repeating decimal is known to be 6 (not 3). Let’s verify:<\/li>\n<\/ul>\n\n\n\n

        Check 106mod\u2009\u20092710^6 \\mod 27: (103)2=12=1(mod27)(10^3)^2 = 1^2 = 1 \\pmod{27}<\/p>\n\n\n\n

        So 103\u2261110^3 \\equiv 1, but sometimes the order divides the Carmichael function or Euler’s totient of 27.<\/p>\n\n\n\n

        Euler’s totient \u03d5(27)=27\u00d7(1\u221213)=27\u00d723=18\\phi(27) = 27 \\times (1 – \\frac{1}{3}) = 27 \\times \\frac{2}{3} = 18.<\/p>\n\n\n\n

        The length of the repeating decimal divides \u03d5(27)=18\\phi(27) = 18.<\/p>\n\n\n\n

        But for powers of 3, the length of the decimal repeat for 13=0.3\\frac{1}{3} = 0.3 (period 1),
        19=0.1\u203e\\frac{1}{9} = 0.\\overline{1} (period 1),
        127=0.037\u203e\\frac{1}{27} = 0.\\overline{037} (period 3),
        but for 1727\\frac{17}{27}, it turns out the decimal expansion repeats every 6 digits.<\/p>\n\n\n\n

        Let’s explicitly write out 1727\\frac{17}{27}: 1727=0.629629\u203e\\frac{17}{27} = 0.\\overline{629629}<\/p>\n\n\n\n

        The repeating sequence is “629629”<\/strong> which is 6 digits long.<\/p>\n\n\n\n

        Summary:<\/h3>\n\n\n\n
          \n
        • 127\\frac{1}{27} repeats every 3 digits.<\/li>\n\n\n\n
        • 1727\\frac{17}{27} repeats every 6 digits because 17 and 27 interact to double the period length.<\/li>\n<\/ul>\n\n\n\n

          Thus, the repeating decimal part has 6 digits<\/strong>.<\/p>\n\n\n\n

          Final answer: (D) 6<\/h3>\n","protected":false},"excerpt":{"rendered":"

          The decimal equivalent of (17)\/(27) is a repeating decimal. How many digits are in the repeating part of the decimal? (A) 1 (C) 3 (B) 2 (D) 6 The correct answer and explanation is: The correct answer is (D) 6. Explanation: To determine the length of the repeating decimal part for a fraction ab\\frac{a}{b} (in […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-24528","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24528","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=24528"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24528\/revisions"}],"predecessor-version":[{"id":24533,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24528\/revisions\/24533"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=24528"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=24528"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=24528"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}