When a number x is divided by 9, the remainder is 6. When the same number is divided by 21, the remainder is 12. If x lies between 250 and 450, then what is the sum of all possible values of x?

To solve this problem, we need to find all numbers x that satisfy both conditions and lie within the given range.

Step 1: First, let's express the conditions mathematically. The first condition can be written as x = 9a + 6, where a is a positive integer. The second condition can be written as x = 21b + 12, where b is a positive integer.

Step 2: We need to find a common expression for x that satisfies both conditions. Since 9 and 21 are multiples of 3, we can rewrite the expressions as follows: x = 3(3a + 2) = 3(7b + 4).

Step 3: From the above, we can see that 3a + 2 must be equal to 7b + 4. Rearranging, we get 3a - 7b = 2.

Step 4: We need to find all pairs of (a, b) that satisfy this equation. By trial and error, we find that (a, b) = (4, 2), (11, 5), (18, 8), (25, 11) satisfy the equation and give x values within the given range.

Step 5: Substitute these pairs into the equation x = 3(3a + 2) to find the possible values of x. For (a, b) = (4, 2), x = 3(34 + 2) = 42. For (a, b) = (11, 5), x = 3(311 + 2) = 99. For (a, b) = (18, 8), x = 3(318 + 2) = 156. For (a, b) = (25, 11), x = 3(325 + 2) = 213.

Step 6: Add up all the possible values of x to get the sum. The sum of all possible values of x is 42 + 99 + 156 + 213 = 510.

So, the sum of all possible values of x that satisfy both conditions and lie within the given range is 510.

To solve this problem, we need to find all numbers x that satisfy both conditions and lie within the given range.

Step 1: First, let's express the conditions mathematically. The first condition can be written as x = 9a + 6, where a is a positive integer. The second condition can be written as x = 21b + 12, where b is a positive integer.

Step 2: We need to find a common expression for x that satisfies both conditions. Since 9 and 21 are multiples of 3, we can rewrite the expressions as follows: x = 3(3a + 2) = 3(7b + 4).

Step 3: From the above, we can see that 3a + 2 must be equal to 7b + 4. Rearranging, we get 3a - 7b = 2.

Step 4: We need to find all pairs of (a, b) that satisfy this equation. By trial and error, we find that (a, b) = (4, 2), (11, 5), (18, 8), (25, 11) satisfy the equation and give x values within the given range.

Step 5: Substitute these pairs into the equation x = 3(3a + 2) to find the possible values of x. For (a, b) = (4, 2), x = 3(34 + 2) = 42. For (a, b) = (11, 5), x = 3(311 + 2) = 99. For (a, b) = (18, 8), x = 3(318 + 2) = 156. For (a, b) = (25, 11), x = 3(325 + 2) = 213.

Step 6: Add all the possible values of x to find the sum. The sum of all possible values of x is 42 + 99 + 156 + 213 = 510.

So, the sum of all possible values of x that satisfy both conditions and lie within the given range is 510.