Find the value of x such that the four-digit number x15x is divisible by 18.
6156 mod 18 = 0, so x = 6.
To find the value of x such that the four-digit number x15x is divisible by 18, we need to determine the divisibility condition for 18.
A number is divisible by 18 if it is divisible by both 9 and 2.
Let's break down the criteria for divisibility by 9 and 2.
1. Divisibility by 9:
A number is divisible by 9 if the sum of its digits is divisible by 9.
2. Divisibility by 2:
A number is divisible by 2 if its last digit is even (ends with 0, 2, 4, 6, or 8).
Now, let's apply these criteria to the given four-digit number x15x.
1. Divisibility by 9:
The sum of the digits in x15x is x + 1 + 5 + x = 2x + 6.
For this number to be divisible by 9, 2x + 6 must be divisible by 9.
To find the possible values of x, we can test different values:
Let's start with x = 0:
2(0) + 6 = 6, which is not divisible by 9.
Trying x = 1:
2(1) + 6 = 8, which is not divisible by 9.
Continue this process for other possible values of x.
2. Divisibility by 2:
The last digit of x15x is x, which can be any even digit (0, 2, 4, 6, or 8), as long as the number remains a four-digit number.
Combining both criteria, the possible values of x that make x15x divisible by 18 are x = 2, 4, 6, or 8.
Therefore, the value of x can be 2, 4, 6, or 8.
To find the value of x such that the four-digit number x15x is divisible by 18, we need to determine the divisibility rule for 18 and then apply it to the given number.
The divisibility rule for 18 states that a number is divisible by 18 if it is divisible by both 9 and 2. In other words, the sum of the digits of the number must be divisible by 9, and the last digit of the number must be even (i.e., divisible by 2).
Let's apply this rule to the given number x15x:
1. Sum of the digits:
The sum of the digits in the number x15x is x + 1 + 5 + x. This can be written as 2x + 6. So, for the number to be divisible by 9, the sum 2x + 6 must be divisible by 9.
2. Last digit:
We have the last digit as x, which can be any digit from 0 to 9. To be divisible by 2, the last digit must be even, which means it can only be 0, 2, 4, 6, or 8.
Combining the two conditions, we need to find the value of x that makes the sum 2x + 6 divisible by 9, and the value of x that is an even number (0, 2, 4, 6, or 8).
Now, for 2x + 6 to be divisible by 9, the possible values of x can be found by considering the remainder when 2x is divided by 9.
By trying different values for x (0, 1, 2, ..., 9), we can find a value that makes 2x + 6 divisible by 9. Starting with x = 0, we solve the equation:
2(0) + 6 = 6 (remainder is 6 when divided by 9)
Next, try x = 1:
2(1) + 6 = 8 (remainder is 8 when divided by 9)
Continue this process until we find a value of x for which 2x + 6 is divisible by 9.
For x = 4, we have:
2(4) + 6 = 14 (remainder is 5 when divided by 9)
For x = 7, we have:
2(7) + 6 = 20 (remainder is 2 when divided by 9)
Thus, none of the values x = 0, 1, 4, or 7 make 2x + 6 divisible by 9.
Therefore, there is no value of x that makes x15x divisible by 18.