What is an odd three-digit number that is divisible by a perfect square and 47?
To find an odd three-digit number that is divisible by both a perfect square and 47, we can follow these steps:
Step 1: Find a perfect square that is a divisor of 47.
We know that 47 is a prime number, and prime numbers are only divisible by 1 and themselves. Since there are no perfect squares less than 47 that are divisors of 47, we move on to the next step.
Step 2: Find a perfect square that is a divisor of a three-digit number.
To find a perfect square that is a divisor of a three-digit number, we can take the square root of the largest three-digit number, which is 999. The square root of 999 is approximately 31.6, so let's choose the nearest perfect square less than or equal to 31, which is 25.
Step 3: Find a common multiple of the perfect square and 47.
To find a common multiple of 25 and 47, we can multiply them together: 25 * 47 = 1175.
Step 4: Check if the resulting number is odd and has three digits.
The resulting number, 1175, is odd since it does not end in 0, 2, 4, 6, or 8. Additionally, it has four digits, not three, so it does not meet the required conditions.
Step 5: Repeat the process if necessary.
Since we did not find a suitable number in Step 4, we need to repeat the process with a different perfect square. The next perfect square less than or equal to 31 is 16.
Step 6: Find a common multiple of the new perfect square and 47.
To find a common multiple of 16 and 47, we multiply them together: 16 * 47 = 752.
Step 7: Check if the resulting number is odd and has three digits.
The resulting number, 752, is even since it ends in 2. It does not meet the requirement of being an odd number.
Step 8: Repeat the process if necessary.
Since we haven't found a suitable number yet, we need to repeat the process with a different perfect square. The next perfect square less than or equal to 31 is 9.
Step 9: Find a common multiple of the new perfect square and 47.
To find a common multiple of 9 and 47, we multiply them together: 9 * 47 = 423.
Step 10: Check if the resulting number is odd and has three digits.
The resulting number, 423, is odd since it does not end in 0, 2, 4, 6, or 8. Additionally, it has three digits, satisfying the required conditions.
Therefore, an odd three-digit number that is divisible by a perfect square (9) and 47 is 423.