Question

Find out the secret number! Here's the clues

It is an integer.
It is a multiple of three, five, and seven.
No digit occurs more than once.
The digit in the tens place is a square number.
The digit in the hundreds place is a cube. The digit in the hundred thousands place is both a square and a cube.
Only the digit in the hundreds place is larger than the digit in the ones place
What is the number?
please show how you got the number please :)​

Answer 1

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!