Grades 11-12 Video Solutions 2023-2023_11-12

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form a, b, b, c, d, d, c, d, d, a, b, b, where the digits a, b, c, and d are themselves four consecutive numbers in some order. What is the value of the digit d?  Alright, so the first thing that we know is if there are six consecutive integers, at least one of them is a multiple of five, at least three of them are a multiple of two.  And that means that the 12-digit number that we have here is actually divisible by both five and two, and so the last digit must be a zero since the number is then divisible by ten.  So that tells us that b is equal to zero. Let's replace all the occurrences of b with zero to simplify the problem. Now our number is a, 0, 0, c, d, d, c, d, d, a, 0, 0.  Since the digits are consecutive, a, c, and d must be one, two, and three, but not necessarily in that order. And their sum is six.  Now one thing to note is that two times d is actually less than or equal to six, and we'll see why this is useful later on.  Among the six consecutive numbers, we must also have at least two multiples of three since there's at least six numbers and six is equal to three times two.  So the 12-digit number must be divisible by nine. For a number to be divisible by nine, the sum of all the digits must also be divisible by nine.  So 2a plus 2c plus 4d must be divisible by nine. Notice that b doesn't matter because it was equal to zero. So that doesn't change the sum.  Since 2a, 2c, and 4d are all even, the sum must also be even. So 2a, 2c, and 4d, their sum must be a multiple of 18.  Figure out the two and we get that 2a plus 2c plus 4d is equal to two times a plus c plus d plus 2d, which is equal to 12 plus 2d.  And now earlier we had noticed that 2d must be less than or equal to six, and so the sum must be equal to 18. It can't be equal to 36. And that only happens if d is equal to three.  So d is equal to three. You can figure out the values of a and c from there, and the correct answer is d is three. That's all we need.  We don't even need the values of a and c, and so our correct answer is c. And we're done.

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