{"id":57761,"date":"2025-08-28T17:05:40","date_gmt":"2025-08-28T11:35:40","guid":{"rendered":"https:\/\/e-gmat.com\/blogs\/?p=57761"},"modified":"2025-09-04T19:46:58","modified_gmt":"2025-09-04T14:16:58","slug":"the-product-of-3305-and-the-1-digit-integer-x-is-a-5-digit-integer","status":"publish","type":"post","link":"https:\/\/e-gmat.com\/blogs\/the-product-of-3305-and-the-1-digit-integer-x-is-a-5-digit-integer\/","title":{"rendered":"The product of 3305 and the 1-digit integer x is a 5-digit integer. The units (ones) digit…."},"content":{"rendered":"A <\/span> 4<\/span> min read <\/span><\/span>\n

Question:<\/strong> The product of 3305 and the 1-digit integer x is a 5-digit integer. The units (ones) digit of the product is 5 and the hundreds digit is y. If A is the set of all possible values of x and B is the set of all possible values of y, then which of the following gives the numbers of A and B?<\/p>\n\n\n\n

Options:<\/p>\n\n\n\n

A. {1, 3, 5, 7, 9} -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}<\/p>\n\n\n\n

B. {1, 3, 5, 7, 9} -> {1, 3, 5, 7, 9}<\/p>\n\n\n\n

c. {3,5,7,9} -> {1,5,7,9}<\/p>\n\n\n\n

D. {5,7,9} -> {1,5,7}<\/p>\n\n\n\n

E. {5,7,9} -> {1,5,9}<\/p>\n\n\n\n

Solution Section<\/h2>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

  • Translate the problem requirements: We need to find all 1-digit values of x where 3305 \u00d7 x gives a 5-digit number with units digit 5, then find all possible hundreds digits y in these products. The sets A and B contain these possible values.<\/li>
  • Apply the 5-digit constraint: Determine which values of x make 3305 \u00d7 x fall between 10,000 and 99,999 to ensure it’s exactly 5 digits.<\/li>
  • Use units digit multiplication pattern: Since 3305 ends in 5, find which single digits x make the units digit of 3305 \u00d7 x equal to 5.<\/li>
  • Calculate actual products and extract hundreds digits: For valid x values, compute 3305 \u00d7 x and identify the hundreds digit in each result to form set B.<\/li><\/ul>\n\n\n\n

    Execution of Strategic Approach<\/h3>\n\n\n\n
    1. Translate the problem requirements<\/strong><\/li><\/ol>\n\n\n\n

      Let’s break down what this problem is asking us to find in everyday terms:<\/p>\n\n\n\n

      • We have the number 3305 and we’re multiplying it by some single digit (1, 2, 3, 4, 5, 6, 7, 8, or 9)<\/li>
      • The result must be a 5-digit number (between 10,000 and 99,999)<\/li>
      • The last digit (units digit) of our result must be 5<\/li>
      • We need to find all possible single digits x that work (this becomes set A)<\/li>
      • For each valid x, we look at the hundreds digit of the result (this becomes set B)<\/li><\/ul>\n\n\n\n

        Think of it like this: if 3305 \u00d7 x = abcde (where abcde is a 5-digit number), then e = 5, and we want to collect all possible values of c (the hundreds digit).<\/p>\n\n\n\n

        Process Skill: TRANSLATE<\/p>\n\n\n\n

        1. Apply the 5-digit constraint<\/strong><\/li><\/ol>\n\n\n\n

          For the product to be exactly 5 digits, it must be at least 10,000 but less than 100,000.<\/p>\n\n\n\n

          Let’s find the range of x values:<\/p>\n\n\n\n

          • Smallest 5-digit number: 10,000 \u00f7 3305 \u2248 3.02<\/li>
          • Largest 5-digit number: 99,999 \u00f7 3305 \u2248 30.25<\/li><\/ul>\n\n\n\n

            Since x must be a single digit (1-9), we need x \u2265 4 to get at least a 5-digit result. Let’s check:<\/p>\n\n\n\n

            • 3305 \u00d7 3 = 9,915 (only 4 digits – too small)<\/li>
            • 3305 \u00d7 4 = 13,220 (5 digits – good!)<\/li><\/ul>\n\n\n\n

              So x must be at least 4, and since x is a single digit, x can be 4, 5, 6, 7, 8, or 9.<\/p>\n\n\n\n

              Process Skill: APPLY CONSTRAINTS<\/p>\n\n\n\n

              1. Use units digit multiplication pattern<\/strong><\/li><\/ol>\n\n\n\n

                Now we need the units digit of the product to be 5. Since 3305 ends in 5, we need to find which single digits, when multiplied by 5, give a units digit of 5.<\/p>\n\n\n\n

                Let’s check the multiplication table for 5:<\/p>\n\n\n\n

                • 1 \u00d7 5 = 5 (units digit is 5 \u2713)<\/li>
                • 2 \u00d7 5 = 10 (units digit is 0 \u2717)<\/li>
                • 3 \u00d7 5 = 15 (units digit is 5 \u2713)<\/li>
                • 4 \u00d7 5 = 20 (units digit is 0 \u2717)<\/li>
                • 5 \u00d7 5 = 25 (units digit is 5 \u2713)<\/li>
                • 6 \u00d7 5 = 30 (units digit is 0 \u2717)<\/li>
                • 7 \u00d7 5 = 35 (units digit is 5 \u2713)<\/li>
                • 8 \u00d7 5 = 40 (units digit is 0 \u2717)<\/li>
                • 9 \u00d7 5 = 45 (units digit is 5 \u2713)<\/li><\/ul>\n\n\n\n

                  So the units digit is 5 when x is odd: 1, 3, 5, 7, or 9.<\/p>\n\n\n\n

                  Combining with our 5-digit constraint (x \u2265 4), we get: x can be 5, 7, or 9.<\/p>\n\n\n\n

                  Wait, let’s double-check x = 3: 3305 \u00d7 3 = 9,915 (only 4 digits), so x = 3 doesn’t work.<\/p>\n\n\n\n

                  Therefore, set A = {5, 7, 9}.<\/p>\n\n\n\n

                  Process Skill: CONSIDER ALL CASES<\/p>\n\n\n\n

                  1. Calculate actual products and extract hundreds digits<\/strong><\/li><\/ol>\n\n\n\n

                    Now let’s calculate the actual products for x = 5, 7, and 9, and find the hundreds digit in each:<\/p>\n\n\n\n

                    • 3305 \u00d7 5 = 16,525 Breaking this down: 1-6-5-2-5 (ten thousands, thousands, hundreds, tens, units) The hundreds digit is 5<\/li>
                    • 3305 \u00d7 7 = 23,135 Breaking this down: 2-3-1-3-5 The hundreds digit is 1<\/li>
                    • 3305 \u00d7 9 = 29,745 Breaking this down: 2-9-7-4-5 The hundreds digit is 7<\/li><\/ul>\n\n\n\n

                      Therefore, set B = {1, 5, 7}.<\/p>\n\n\n\n

                      1. Final Answer<\/strong><\/li><\/ol>\n\n\n\n

                        We found:<\/p>\n\n\n\n

                        • Set A (possible values of x) = {5, 7, 9}<\/li>
                        • Set B (possible values of y, the hundreds digit) = {1, 5, 7}<\/li><\/ul>\n\n\n\n

                          Looking at the answer choices, this matches option (D): {5, 7, 9}———-{1, 5, 7}<\/p>\n\n\n\n

                          Answer: D<\/p>\n\n\n\n

                          Common Faltering Points<\/h3>\n\n\n\n

                          Errors while devising the approach<\/strong><\/p>\n\n\n\n

                          • Missing the 5-digit constraint: Students often focus only on finding values of x where the units digit is 5, but forget that the product must specifically be a 5-digit number. This leads them to include x = 1 and x = 3 in set A, even though 3305 \u00d7 1 = 3,305 (4 digits) and 3305 \u00d7 3 = 9,915 (4 digits) don’t meet the requirement.<\/li>
                          • Misunderstanding what the hundreds digit means: Students may confuse which digit position represents the hundreds place in a 5-digit number. For a number like 23,135, they might incorrectly identify 3 (thousands digit) or 3 (tens digit) as the hundreds digit instead of the correct digit 1.<\/li>
                          • Overlooking the constraint that x must be a 1-digit integer: Some students might consider x = 0 as a possibility, not realizing that multiplying by 0 would give a product of 0, which is not a 5-digit number, or they might not restrict themselves to single digits 1-9.<\/li><\/ul>\n\n\n\n

                            Errors while executing the approach<\/strong><\/p>\n\n\n\n

                            • Arithmetic errors in multiplication: Students may make calculation mistakes when computing products like 3305 \u00d7 7 = 23,135 or 3305 \u00d7 9 = 29,745. Even small errors here will lead to wrong hundreds digits for set B.<\/li>
                            • Incorrect boundary calculations: When determining the range for x to ensure a 5-digit result, students might make errors in division (like 10,000 \u00f7 3305) or incorrectly round the results, leading them to include or exclude wrong values of x.<\/li>
                            • Units digit pattern errors: Students might incorrectly determine which values of x give a units digit of 5 when multiplied by 5, possibly confusing the pattern or making errors in the multiplication table for 5.<\/li><\/ul>\n\n\n\n

                              Errors while selecting the answer<\/strong><\/p>\n\n\n\n

                              • Mixing up sets A and B: Students might correctly find the values {5, 7, 9} and {1, 5, 7} but accidentally reverse which set corresponds to A (possible x values) versus B (possible hundreds digits), leading them to select an incorrect answer choice.<\/li>
                              • Including invalid x values in the final answer: Even after discovering that x = 1 and x = 3 don’t produce 5-digit numbers, students might still include these in their final set A, selecting answer choice (B) instead of the correct (D).<\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"

                                Question: The product of 3305 and the 1-digit integer x is a 5-digit integer. The units (ones) digit of the product is 5 and the hundreds digit is y. If A is the set of all possible values of x and B is the set of all possible values of y, then which of the […]<\/p>\n","protected":false},"author":102457,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"off","_et_pb_old_content":"","_et_gb_content_width":"","ub_ctt_via":""},"categories":[44,100],"tags":[],"featured_image_src":null,"author_info":{"display_name":"Kashish Garg","author_link":"https:\/\/e-gmat.com\/blogs\/author\/kashish\/"},"yoast_head":"\nThe product of 3305 and the 1-digit integer x is a 5-digit integer. The units (ones) digit....<\/title>\n<meta name=\"description\" content=\"The product of 3305 and the 1-digit integer x is a 5-digit integer. 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