Answer:
a) 102 is the smallest of the dropdown number
b) 1025 is the smallest dropdown number
c) 9874 is the largest 4-digit dropdown number
Step-by-step explanation:
Given:
All digits in a dropdown number are different and one of its digits is the average of all its digits.
It has at least two digits.
To find:
a) the smallest dropdown number
b) the smallest and largest 4-digit dropdown numbers
a) From the information given, the numbers are at least two digits. This means the smallest of the digits from the dropdown is 2
We can't have 0 as the starting number because it will reduce the digits by 1. The list will start from the smallest to the highest since we are looking for the smallest number.
For two digits: 12, 13, 14, etc
We can only have a 2-digit number whose average is part of digits if the numbers are the same digit.
As a result, we will move to 3 digits.
For 3 dgits: 102, 103, 104, 105, etc
We will check if any of the above will have one of its digits as average
102: 1 + 0 + 2 = 3
Average = 3/3 = 1
One of the digits of this number is the average of the number.
Hence, 102 is the smallest of the dropdown number
b) To get the smallest 4 digit dropdown number, we will list from smallest to highest
For 4 digits: 1023, 1024, 1025, 1026, 1027, ...
We will check if any of the above will have one of its digits as average
1023: 1+0+2+3 = 6
Average = 6/4 = 1.5
1024: 1+0+2+4 = 7
Average = 7/4 = 1.75
1025: 1+0+2+5 = 8
Average = 8/4 = 2
One of the digits of this number is the average of the number.
Hence, 1025 is the smallest dropdown number
To get the largest 4-digit dropdown number, we will list from largest to smallest
For 4 digits: 9876, 9875, 9874, 9873, 9872, etc
We will check if any of the above will have one of its digits as average
9876: 9+8+7+6 = 30
Average = 30/4 = 7.5
9875: 9+8+7+5 = 29
Average = 29/4 = 7.25
9874: 9+8+7+4 = 28
Average = 28/4 = 7
One of the digits of this number is the average of the number.
Hence, 9874 is the largest 4-digit dropdown number