Rambling on and on
Above shows an incomplete long multiplication with each boxes represent a distinct single digit positive integer other than 2. Find the sum of values of all the missing numbers.
The answer is 35.
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Since each boxes represent a distinct single digit positive integer that isn't 2, the top box in the right column:
cannot be 1, because the last digit (the bottom box in the last column) will be 1 x 2 = 2.
cannot be 2 as well.
cannot be 5, because the last digit will be 5 x 2 mod 10 = 0, which is not a positive integer.
cannot be 6, because the last digit will be 6 x 2 mod 10 = 2.
Hence, that leaves us with 23, 24, 27, 28, and 29 for the number on top.
Let the digit of the top box in the centre column be n. We know that (10n + 2) times the number on top must be greater than 2000. We know that n cannot be 1, 2, 3, 4, 5, or 6, because the greatest product that could be produced by multiplying the number below with the number on top is 62 x 29 = 1998, which is less than 2000.
Hence, that leaves us with 72, 82, and 92 for the number below.
Since there are only 5 x 3 = 15 possible combinations, we can use the trial and error method to find out which of the combinations would satisfy the given condition. We need to keep in mind that other than the digit '2', no other digit should occur more than once in the line of each multiplication together with its product and that the digit '2' should only appear 3 times. '0' should not appear either because it is not a positive integer.
23 x 72 = 1656 (6 is repeated)
23 x 82 = 1886 (8 is repeated)
23 x 92 = 2116 (1 is repeated)
24 x 72 = 1728 (7 is repeated)
24 x 82 = Notice that the last digit will be 4 x 2 = 8, which is already used as a part of the multipliers and hence we do not need to calculate the result to know that this combination does not satisfy the question.
24 x 92 = 2208 (there is a 0)
27 x 72 = No need to calculate since 7 is already repeated in the multipliers.
27 x 82 = 2214 (2 appears more than thrice)
27 x 92 = 2484 (4 is repeated)
28 x 72 = 2016 (there is a 0)
28 x 82 = 2296 (2 appears more than thrice)
28 x 92 = 2576 (satisfies the given conditions)
Even though there is already a combination that satisfies the given conditions, there is no hurt checking for other possible solutions, right?
29 x 72 = 2088 (there is a 0)
29 x 82 = Notice that the last digit will be 4 x 2 = 8, which is already used as a part of the multipliers and hence we do not need to calculate the result to know that this combination does not satisfy the question.
29 x 92 = No need to calculate since 9 is already repeated in the multipliers.
Therefore, we can conclude that 28 x 92 = 2576 is the one and only combination that satisfies all the given conditions. Hence, the sum of the missing numbers is 8 + 9 + 5 + 7 + 6 = 35