I am a 3digit number divisible by 3. my tens digit is three times as great as my hundreds digit and the sum of my digits is 15. if you were reverse my digits I am divisible by six, as well as by 3. what number am I?
Let's break down the information given:
1. The number is divisible by 3.
2. The tens digit is three times as great as the hundreds digit.
3. The sum of the digits is 15.
4. If the digits are reversed, the number is divisible by both 3 and 6.
Let's start by finding the possible values for the hundreds digit. Since the sum of the digits is 15, and the tens digit is three times the hundreds digit, the possible values for the hundreds digit are 1, 2, 3, and 4.
Let's go through each possible value of the hundreds digit one by one:
1. If the hundreds digit is 1, then the tens digit would be 3 times 1, which is 3. The sum of the digits is 15, so the units digit would be 15 - 1 - 3 = 11. However, this is not a possible value for the units digit, as it exceeds 9 (since the number is 3 digits long). Therefore, 1 is not a possible hundreds digit.
2. If the hundreds digit is 2, then the tens digit would be 3 times 2, which is 6. The sum of the digits is 15, so the units digit would be 15 - 2 - 6 = 7. Therefore, if the hundreds digit is 2, the number would be 267.
3. If the hundreds digit is 3, then the tens digit would be 3 times 3, which is 9. The sum of the digits is 15, so the units digit would be 15 - 3 - 9 = 3. However, this is not a possible value for the units digit, as it does not satisfy the condition that the tens digit is three times the hundreds digit. Therefore, 3 is not a possible hundreds digit.
4. If the hundreds digit is 4, then the tens digit would be 3 times 4, which is 12. The sum of the digits is 15, so the units digit would be 15 - 4 - 12 = -1. However, this is not a possible value for the units digit, as the units digit must be a non-negative number. Therefore, 4 is not a possible hundreds digit.
Taking into account all the possible values for the hundreds digit, we find that the only solution that satisfies all the given conditions is the number 267.
To find the number that satisfies all the given conditions, we can set up the following steps:
Step 1: Determine the possible values for the hundreds digit.
Since the number is divisible by 3 and the sum of the digits is 15, the possible values for the hundreds digit are 1, 4, and 7. (These values ensure that when the hundreds digit is multiplied by 3, the result is a single-digit number.)
Step 2: Determine the possible values for the tens digit.
The tens digit is three times as great as the hundreds digit. We can try all possible values for the hundreds digit (1, 4, and 7) and calculate the corresponding tens digit. For example, if the hundreds digit is 1, then the tens digit is 3 x 1 = 3. If the hundreds digit is 4, then the tens digit is 3 x 4 = 12. If the hundreds digit is 7, then the tens digit is 3 x 7 = 21.
However, since the tens digit must be a single-digit number, we can eliminate the value 21.
Step 3: Determine the possible values for the units (ones) digit.
The units digit can be calculated by subtracting the sum of the hundreds and tens digits from 15. For example, if the hundreds digit is 1 and the tens digit is 3, then the units digit is 15 - 1 - 3 = 11. If the hundreds digit is 4 and the tens digit is 12, then the units digit is 15 - 4 - 12 = -1. However, since we are dealing with positive integers, we cannot have a negative value for the units digit. Therefore, we can eliminate the values 11 and -1.
Step 4: Determine the number that satisfies all the conditions.
By following the steps above, we found that the hundreds digit can be 1 or 4, and the tens digit can be 3 or 12. Let's check which combination satisfies the final condition of being divisible by 6 when the digits are reversed.
If the hundreds digit is 1 and the tens digit is 3, the number would be 113. However, 311 is not divisible by 6. Thus, this combination is not valid.
If the hundreds digit is 4 and the tens digit is 12, the number would be 412. Reversing the digits gives us 214, which is divisible by 6.
Therefore, the number that satisfies all the given conditions is 412.