Answer:
Explanation:
Known facts:
- multiple of three, five, and seven
- No digit occurs more than once
- digit in the tens place is a square number.
- the digit in the hundreds place is a cube
- The digit in the hundred thousand places is both a square and a cube.
- Only the digit in the hundreds place is larger than the digit in the ones place
Analyzing our choices:
- since the tens place is a square, but cannot be larger than the ones place, it has to be either 1 or 4
- the hundreds place is a cube which can be 8 not one since the digit in the hundred thousand places is both a square and a cube which is 1
- also, the number is divisible by 5 meaning that the last digit must be 5 or zero. However, only the hundreds place is larger than the digit in the ones place so it must be 5
Looking at what we have so far:
1 _ _ 8 4 5
- this number is divisible by 3 and cannot be larger than 5 which is in the units place
- when a number is divisible by 3 all its digits must add to a number that is divisible by 3, meaning that the last unfilled digits can be filled by '0' and '3'
- now the order of '0' and '3' is determined by whether the number is divisible by 7
Result: 103845 is the secret number
Hope that helps!