How many numbers ¯abcd with different digits satisfy the following property: if we replace the largest digit with the digit 1 results in a multiple of 30?
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Tags: number theory, Digits, multiple, divisible
22.09.2021 23:35
For the resulting number to be a multiple of 30, it must be a multiple of 3 and 10. To be divisible by 10, the last digit must be 0 Hence d is 0. Note that we can't substitute the digit d otherwise the units digit will be the digit 1. One of the numbers from {a,b,c} will be 1 so the remaining two numbers k and m must satisfy k+m \equiv 2 (\mod 3). The possible remaining k and m values are: (8, 6), (8, 3), (7, 4), (7, 1), (6, 5), (6, 2), (5, 3), (4, 1), (3, 2). There are then a total of 27 unordered sets of the remaining digits. where the value substituted has to be greater than k and m. Furthermore, there are 3! ways of arranging these digits. Hence, the answer is 6 \cdot 27 = \boxed{162}
22.09.2021 23:48
Without restrictions, there are a total of 9\cdot 9 \cdot 8 \cdot 7=4536 combinations. In order for the number to be a multiple of 30, it must be divisible by 10 and 3, in which the restriction that it is divisible by 10 will make d be equal to 0. Fixing 0 to d make the total combinations equal to 9\cdot 8 \cdot 7 \cdot 1=504 combinations. Note that one number must be 1.Now, we must restrict the number to be a multiple of 3. The divisibility rule of 3 states that the sum of the digits must be divisible by 3 in order to be divisible such that the remaining two digits b and c satisfy the following: b+c=5, b+c=8, b+c=11, b+c=14, and b+c=17 are the possible sums. The values of b and c will be in the ordered pairs of (3,2), (4,1), (5,3), (6,2), (6,5), (7,1), (7,4), (8,3), and (8,6). I believe there are a total of 27 sets of the remaining digits in which the Replaced value, x, is x>b and x>c. Also, there a total of 6 ways to arrange the digits; Hence the answer is 27 \cdot 6, or 162 numbers that satisfy the conditions.
22.09.2021 23:49
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22.09.2021 23:49
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23.09.2021 00:02
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23.09.2021 03:44
We are at the skewl