Find the mystery number
im a number between 300 and 400
im divisible by 3 and 4 . but not 9
my tens digit is a square number
the sum of the hundreds digit and the tens digit is a prime number.
find the mystery number
we know that the number is 3xx
the tens digit is either 1,4, or 9
4+3 =7 is a prime, so the number is 34x
x cannot be 2 or 5, because 342 and 345 are divisible by 9 (cast out nines)
The number must be even, because it is divisible by 4. So, it is either
340 344 346 348 (but 4 does not divide 346)
Only 348 is divisible by 3 (cast out threes)
To find the mystery number, let's follow these steps:
Step 1: Start with the range between 300 and 400.
Step 2: Look for numbers within this range that are divisible by both 3 and 4.
Step 3: Check if the number is not divisible by 9.
Step 4: Identify the numbers whose tens digit is a square number. The possible square numbers for the tens digit in this range are 1, 4, and 9.
Step 5: Find the numbers where the sum of the hundreds digit and the tens digit is a prime number.
Let's apply these steps to find the mystery number:
Between 300 and 400, the numbers divisible by both 3 and 4 are 312, 324, 336, 348, and 372.
Among these numbers, let's check if any of them are divisible by 9:
- 312 is not divisible by 9 since the sum of its digits (3 + 1 + 2 = 6) is not divisible by 9.
- 324 is divisible by 9 since the sum of its digits (3 + 2 + 4 = 9) is divisible by 9.
Since we know that the mystery number is not divisible by 9, we can eliminate 324 from the list.
Next, let's check the tens digits of the remaining numbers to find if they are square numbers:
- 312's tens digit is 1, which is a square number.
- 336's tens digit is 3, which is not a square number.
- 348's tens digit is 4, which is a square number.
- 372's tens digit is 7, which is not a square number.
Now, let's find the numbers where the sum of the hundreds digit and the tens digit is a prime number:
- For 312, the sum of the hundreds digit (3) and the tens digit (1) is 4, which is not a prime number.
- For 348, the sum of the hundreds digit (3) and the tens digit (4) is 7, which is a prime number.
Therefore, the mystery number is 348.
To find the mystery number, we will break down the given clues step by step.
1. The number is between 300 and 400:
This means the mystery number is greater than or equal to 300, and less than 400.
2. The number is divisible by 3 and 4, but not 9:
To find a number that is divisible by both 3 and 4, we need to find a number that is also divisible by their least common multiple, which is 12. So, the mystery number must be divisible by 12. We can start with the number 312 since it falls within the given range and is divisible by 12.
3. The tens digit is a square number:
A square number is a number multiplied by itself. The possible square numbers for the tens digit are 1, 4, and 9. Checking these possibilities, we find that the mystery number cannot be 312, as 312 does not have a square number as its tens digit.
4. The sum of the hundreds digit and the tens digit is a prime number:
For this clue, we need to check all possible combinations of the hundreds and tens digits to find a prime sum. The prime numbers less than 10 are 2, 3, 5, and 7.
Trying the combinations (3, 4) and (4, 3), we find that the hundreds digit cannot be 3. Therefore, the sum of the hundreds digit and the tens digit must be 7. The only remaining possibility is (2, 5), so the mystery number is 325.
Therefore, the mystery number is 325.