This is a riddle, and I'm supposed to find a 5-digit number from it.....
My thousand's digit is the first that is perfect,
but if you're not happy with that,
it's also the product of the first two primes.
My units digit is the odd of those two primes,
and if you look at it and my tens,
you'll find a number divisble by three, but not by the square of that my friends.
Now let's look at my hundreds and thousands
If you examine those two as they're paired, you'll find they form a number divisible by an even prime that's been squared. (4)
Now let's look at my body in whole. Five different digits you see, in order (or not) you'll find that they are divisible by nine,
Can you find the mystery number?
I think I have the thousands and hundreds digits...
Hundreds: 3
Thousands: 6
I'm lost beyond that.
To solve this riddle and find the 5-digit number, let's break down the clues step by step.
1) The thousand's digit is the first that is perfect, but if you're not happy with that, it's also the product of the first two primes.
The first perfect number is 6, as it is equal to the sum of its divisors (1+2+3=6). However, we need to check if it can also be expressed as the product of the first two primes. The first two primes are 2 and 3, and their product is 6. So, the thousand's digit is indeed 6.
2) The units digit is the odd of those two primes, and if you look at it and my tens, you'll find a number divisible by three, but not by the square of that my friends.
The two primes mentioned earlier, 2 and 3, have an odd value of 3. Now we need to find a combination of the units and tens digit where the number is divisible by 3, but not by its square (9). Let's try different combinations until we find the right one.
- 63: Divisible by 3, but 9 divides it.
- 33: Divisible by 3, but 9 divides it.
- 93: Not divisible by 3, so it's not a valid option.
- 13: Not divisible by 3, so it's not a valid option.
- 23: Divisible by 3, and 9 does not divide it.
The units digit is 3, and the tens digit is 2.
3) Now let's look at my hundreds and thousands. If you examine those two as they're paired, you'll find they form a number divisible by an even prime that's been squared (4).
The hundreds and thousands digits are 3 and 6, respectively. We need to create a number with these digits that is divisible by an even prime squared. The only even prime is 2, and when squared, it gives 4. So, the hundreds and thousands digits, when paired, form the number 36, which is divisible by 4.
4) Now let's look at my body in whole. Five different digits you see, in order (or not). You'll find that they are divisible by nine.
We already have the thousands digit (6), hundreds digit (3), tens digit (2), and units digit (3). We just need to find a unique fifth digit that, when added to the previous digits, gives a number divisible by 9.
- 63233: Divisible by 9.
- 63213: Not divisible by 9.
- 63223: Not divisible by 9.
- 63243: Divisible by 9.
The fifth digit is 4.
Assembling all the digits, the mystery number is 63243.
Let's break down the riddle step by step to find the missing digits:
1. My thousand's digit is the first that is perfect, but if you're not happy with that, it's also the product of the first two primes.
The first perfect number is 6 (1 + 2 + 3). So, the thousands digit is 6.
2. My units digit is the odd of those two primes, and if you look at it and my tens, you'll find a number divisible by three, but not by the square of that, my friends.
The first two primes are 2 and 3. The odd one is 3. Now we need to find a number divisible by 3 and has a units/tens digit not divisible by the square root of 3, which is √3 ≈ 1.732. Checking the possibilities, we find that 3 and 6 satisfy this condition.
3. Now let's look at my hundreds and thousands. If you examine those two as they're paired, you'll find they form a number divisible by an even prime that's been squared. (4)
The even prime squared is 4. So, the hundreds digit is 4.
4. Now let's look at my body as a whole. Five different digits you see, in order (or not). You'll find that they are all divisible by nine.
Since there are five different digits, let's assign the remaining digits as 5, 2, and 1. Now we need to arrange them in such a way that the resulting number is divisible by 9. The sum of the five digits is 6 + 4 + 3 + 5 + 2 + 1 = 21, which is divisible by 9. So, the remaining digits are 5, 2, and 1.
Putting all the digits together, we have 64321. Therefore, the mystery number is 64321.