i am a 3 digit number divisible by 3.my tens digit is 3 times as great as my hundreds digit and the sum of my digits is 15. if you reverse my digits i am divisible by 6 and 3...... what number am i?
We know that the number is divisible by three if the sum of its digits is 15!
It's got 3 digits. When reversed, it is divisible by 6, which means that the reversed number is divisible by 2. This translates to
"the hundred's digit is even".
From
"my tens digit is 3 times as great as my hundreds digit"
we deduce that the number must be one of the three possibilities:
A. 13X
B. 26X
C. 39X
Where X can be found from the fact that the digits add up to 15.
Both A and C will not work because the hundreds digit is not even.
So can you find the number now?
Let's solve this step by step:
Step 1: Identify the conditions given in the question.
- The number is a 3-digit number divisible by 3.
- The tens digit is 3 times as great as the hundreds digit.
- The sum of the digits is 15.
Step 2: Start by finding the possible values for the hundreds digit.
- Since the sum of the digits is 15, the hundreds digit must be less than or equal to 7 (since 7 + 7 + 1 = 15)
- Let's try different values for the hundreds digit: 1, 2, 3, ..., 7.
Step 3: Calculate the tens digit based on the given conditions.
- The tens digit is 3 times the value of the hundreds digit.
Let's go through each possibility.
For hundreds digit = 1:
- Tens digit = 3 * 1 = 3
- Units digit = 15 - (1 + 3) = 11 (which is not possible for a single-digit number)
For hundreds digit = 2:
- Tens digit = 3 * 2 = 6
- Units digit = 15 - (2 + 6) = 7
For hundreds digit = 3:
- Tens digit = 3 * 3 = 9
- Units digit = 15 - (3 + 9) = 3
For hundreds digit = 4:
- Tens digit = 3 * 4 = 12 (which is not possible since the tens digit should be a single digit)
For hundreds digit = 5:
- Tens digit = 3 * 5 = 15 (which is not possible since the tens digit should be less than or equal to 9)
For hundreds digit = 6:
- Tens digit = 3 * 6 = 18 (which is not possible since the tens digit should be less than or equal to 9)
For hundreds digit = 7:
- Tens digit = 3 * 7 = 21 (which is not possible since the tens digit should be less than or equal to 9)
So, the only possibility that meets all the given conditions is when the hundreds digit is 3, tens digit is 9, and units digit is 3.
Thus, the number is 393.
To find the number that satisfies all the given conditions, we can proceed step by step:
Step 1: "I am a 3-digit number divisible by 3."
Since we are looking for a number divisible by 3, we know that the sum of its digits must also be divisible by 3.
Step 2: "My tens digit is 3 times as great as my hundreds digit."
Let's denote the hundreds digit as "x" and the tens digit as "3x" since the tens digit is three times greater than the hundreds digit.
Step 3: "The sum of my digits is 15."
We know that the sum of the hundreds digit, tens digit, and units digit is 15. We already have the tens digit "3x," and the units digit will be equal to the remaining value: 15 - (x + 3x).
Step 4: "If you reverse my digits, I am divisible by 6 and 3."
If we reverse the digits, the number will still be divisible by 6 and 3. Divisibility by 6 means the number must be even and divisible by 3, which we have already ensured. Divisibility by 3 is already satisfied by the initial condition.
Now, let's solve for the unknown values:
Step 5: The sum of the digits is 15.
x + 3x + (15 - (x + 3x)) = 15
4x + 15 - 4x = 15
15 = 15
This equation doesn't give us any information about "x" because x cancels out on both sides of the equation.
Step 6: Solve for the hundreds digit, "x."
Since we don't have a unique solution for 'x,' we can try different values. Let's start with x = 3.
When x = 3:
tens digit = 3x = 3(3) = 9
units digit = 15 - (x + 3x) = 15 - (3 + 9) = 15 - 12 = 3
Therefore, the number is 393.
You can also check that the number meets all the conditions:
- Divisible by 3: 393 ÷ 3 = 131 (no remainder)
- Tens digit is 3 times the hundreds digit: 3 × 3 = 9 (tens digit)
- The sum of the digits is 15: 3 + 9 + 3 = 15
- If you reverse the digits, the number is divisible by 6 and 3: 393 ÷ 6 = 65.5 (divisible by 6), 393 ÷ 3 = 131 (divisible by 3).