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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. For example, 5021 is a dropdown number, but neither 4389 nor 6033 is a dropdown number.1. Find the smallest dropdown number2. What are the smallest and largest 4-digit dropdown numbers? Please provide working out for a and b. Thanks.

User Radu Grama
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1 Answer

15 votes
15 votes
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

User Frogb
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