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, as well as by 3.
What number am I?
Thank you for your help.
If the hundreds digit is is 1, the tens must be 3, likewise 2 and 6, 3 and 9.
To get 15, the possibilities exclude 1 and 3, because the ones digit cannot be a single digit.
The remaining possibilities are 267 and 393.
Which one fits all the requirements?
I hope this helps.
To find the 3-digit number that satisfies the given conditions, let's go step by step:
Step 1: Identify the tens and hundreds digits.
Let's assume the hundreds digit is x.
Given that the tens digit is 3 times the value of the hundreds digit: tens digit = 3 * x
Step 2: Determine the sum of the digits.
The sum of the digits is given as 15: x + (3 * x) + (x + (3 * x)) = 15
Simplifying the equation: 8x + 3x = 15
Combining like terms: 11x = 15
Step 3: Solve for x.
Dividing both sides of the equation by 11: x = 15/11
Since the solution for x is not a whole number, the problem does not have a valid solution.
Therefore, there is no 3-digit number that satisfies all the given conditions.
To solve this problem, we need to break it down into steps and use logical reasoning. Let's start by going through the given information step by step.
1. You are a 3-digit number divisible by 3.
Since division by 3 requires the sum of the digits to be divisible by 3, we know that the sum of the digits of the unknown number must be divisible by 3.
2. The tens digit is 3 times as great as the hundreds digit.
Let's represent the hundreds digit with the variable "x" and the tens digit with "3x" since it is three times as great. The units digit will be represented with "y".
So, the unknown number can be written as "xyz".
3. The sum of the digits is 15.
We can sum up the digits using the information we already have:
x + 3x + y = 15
4. Reversing the digits makes the number divisible by 6 and 3.
When we reverse the digits, the new number will be "zyx". For this new number to be divisible by 6 and 3, the sum of the new digits should also be divisible by 3, and the units digit should be an even number.
Based on this information, the new sum of the digits can be represented as:
y + 3x + x
5. Solving the equations simultaneously.
We can now solve the system of equations consisting of the sum of the digits and the reversed digits being divisible by 3 and 6:
x + 3x + y = 15
y + 3x + x is divisible by 6 and 3
Simplifying the second equation:
4x + y
Since the number is divisible by both 3 and 6, the sum of the digits must be a multiple of 3, and the units digit must be even.
Looking at the sum of the digits equation, we have x + 3x + y = 15, which means that 4x + y is also a multiple of 3. The possible values for (4x + y) that are multiples of 3 and even are 6 and 12.
By trying different values, we find a solution: x = 3, y = 6.
So, the unknown number is 336.
To confirm, we can check:
- The sum of the digits (3 + 3 + 6) is 12, which is divisible by 3.
- Reversing the digits gives us 633, which is divisible by 6 and 3.
Therefore, the number you are looking for is 336.