If the number 4x315 is divisible by 3, where x is a digit, what can be the sum of all such values of x?
15
Understanding Divisibility by 3
The question asks us to find the sum of all possible values of the digit 'x' in the number 4x315 such that the number is divisible by 3.
To solve this, we need to recall the divisibility rule for 3. A number is divisible by 3 if and only if the sum of its digits is divisible by 3.
Applying the Divisibility Rule to 4x315
Let's find the sum of the digits in the number 4x315. The digits are 4, x, 3, 1, and 5.
The sum of the digits is:
$$ \text{Sum of digits} = 4 + x + 3 + 1 + 5 $$
Adding the known digits:
$$ 4 + 3 + 1 + 5 = 13 $$
So, the sum of the digits is:
$$ \text{Sum of digits} = 13 + x $$
For the number 4x315 to be divisible by 3, the sum of its digits, which is $13 + x$, must be divisible by 3.
Finding Possible Values of the Digit 'x'
Since 'x' is a digit, its possible values are from 0 to 9 (i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
We need to find which of these values for 'x' make $13 + x$ divisible by 3. Let's test each possible value or think about multiples of 3 greater than or equal to 13.
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
We are looking for values of $13 + x$ that are multiples of 3. Since 'x' is a digit (0-9), the minimum value of $13 + x$ is $13 + 0 = 13$, and the maximum value is $13 + 9 = 22$.
So, the possible multiples of 3 that $13 + x$ can be are 15, 18, and 21.
- If $13 + x = 15$, then $x = 15 - 13 = 2$. Since 2 is a digit, x=2 is a possible value. ($13+2=15$, divisible by 3)
- If $13 + x = 18$, then $x = 18 - 13 = 5$. Since 5 is a digit, x=5 is a possible value. ($13+5=18$, divisible by 3)
- If $13 + x = 21$, then $x = 21 - 13 = 8$. Since 8 is a digit, x=8 is a possible value. ($13+8=21$, divisible by 3)
- If $13 + x = 24$, then $x = 24 - 13 = 11$. Since 11 is not a single digit, this is not a possible value for x.
Therefore, the possible values for the digit 'x' are 2, 5, and 8.
Calculating the Sum of Possible Values of x
The question asks for the sum of all such values of x. The possible values are 2, 5, and 8.
Sum of these values = $2 + 5 + 8$
$$ 2 + 5 + 8 = 15 $$
The sum of all possible values of x is 15.
| Value of x | Sum of digits (13 + x) | Is $13 + x$ divisible by 3? | Is x a valid digit (0-9)? | Is x a possible value? |
|---|---|---|---|---|
| 0 | 13 | No | Yes | No |
| 1 | 14 | No | Yes | No |
| 2 | 15 | Yes | Yes | Yes |
| 3 | 16 | No | Yes | No |
| 4 | 17 | No | Yes | No |
| 5 | 18 | Yes | Yes | Yes |
| 6 | 19 | No | Yes | No |
| 7 | 20 | No | Yes | No |
| 8 | 21 | Yes | Yes | Yes |
| 9 | 22 | No | Yes | No |
The possible values for x are indeed 2, 5, and 8. Their sum is 15.
Revision Table: Key Concepts in Divisibility by 3
| Rule | Explanation | Example |
|---|---|---|
| Sum of Digits | A number is divisible by 3 if the sum of its individual digits is a multiple of 3. | Is 123 divisible by 3? Sum of digits = 1 + 2 + 3 = 6. Since 6 is divisible by 3, 123 is divisible by 3. |
Additional Information: Divisibility Rules
Understanding divisibility rules can help quickly determine if a number is divisible by another number without performing long division. Here are a few common rules:
- Divisibility by 2: A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, 8).
- Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
- Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
- Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. (Similar to the rule for 3, but stricter).
- Divisibility by 10: A number is divisible by 10 if its last digit is 0.
These rules are helpful for solving problems involving number properties and factors.
Similar Questions
What is the least number which when doubled is perfectly divisible by 7, 12 and 15?
When x 2+ ax + b is divided by (x - 1), the remainder is 15 and when x 2+ bx + a is divided by (x + 1), the reminder is -1, then the value of a 2+ b 2is:
Find the smallest square number from among the given options, which is divisible by each of 8, 15 and 20.
If the 8 digit number 136p5785 is divisible by 15, then find the least possible value of P.
How many of the factors of 360 are perfect squares?
Which of the following numbers is divisible by 12?
- If the number x3331 is divisible by 11, what is the face value of x?
1
?
3
Find the missing digit if it has 11 and 13 as factors?
A 4-digit number 1xy7 is divisible by 11. What is the value of x - y?
Which of these numbers is divisible by 6?
Important Questions from Divisibility and Remainder
If the 8-digit number 888x53y4 is divisible by 72, then what is the value of (7x + 2y), for the maximum value of y?
If all positive divisors of 132 are arranged in descending order, then what digit will be at unit place of first divisor ?
If 3 2019 is divided by 10, then what is the remainder?
The number 3798125P369 is divisible by 7. What is the value of the digit P?
Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are multiples of 3. Let S be their sum.
Which of the following is/are correct?
1. S is always divisible by 74.
2. S is always divisible by 9.
select the correct answer using the code given below:
