1. **Problem 1: Conditions for divisibility by 12** Given the number 5aa1a, we need to determine for which digits \(a\) it is divisible by 12. Recall that a number is divisible by 12 if and only if it is divisible by both 3 and 4. - Divisibility by 3: Sum of digits must be divisible by 3. Sum of digits = 5 + a + a + 1 + a = 6 + 3a. This sum must be divisible by 3, so \(6 + 3a\) is divisible by 3. Since 6 is divisible by 3, \(3a\) must be divisible by 3, meaning \(a\) can be any digit (0 through 9). - Divisibility by 4: Last two digits must form a number divisible by 4. The last two digits are \(1a\), i.e., \(10 \times 1 + a = 10 + a\). So \(10 + a\) must be divisible by 4. Checking \(a = 0 \text{ to } 9\): For \(a=2\), \(10+2=12\) divisible by 4. For \(a=6\), \(10+6=16\) divisible by 4. For \(a=10\) not possible since \(a\) is a digit. So possible values are \(a=2,6\). Therefore, the number 5aa1a is divisible by 12 if \(a=2\) or \(a=6\). 2. **Problem 2: Counting three-digit even numbers with digits 2,3,9,5** We want the number of three-digit even numbers with distinct digits formed from \(\{2,3,9,5\}\). - Since the number is even, its last digit must be even. The only even digit here is 2. - So the last digit is fixed as 2. - The first two digits are chosen from \(\{3,9,5\}\) without repetition. - Number of ways for the first digit: 3 choices. - Number of ways for the second digit: 2 choices. Total numbers = \(3 \times 2 = 6\). 3. **Problem 3: Classification of number 5,307** - The number 5,307 has a comma. Assuming it represents five thousand three hundred seven - It is a whole number without decimal component. - Hence it is an integer. - Since it is not a fraction or decimal, it is "اصم" (whole number or integer). So, 5,307 is an integer (اصم). 4. **Problem 4: Find midpoint coordinates** Given segment \(\Delta\) graded on \(0;1\) and points \(A\) and \(B\) with distances 2 and \(-\frac{3}{5}\) respectively. The midpoint \(C\) coordinate is average of coordinates of \(A\) and \(B\): $$ C = \frac{2 + \left(-\frac{3}{5}\right)}{2} = \frac{2 - \frac{3}{5}}{2} = \frac{\frac{10}{5} - \frac{3}{5}}{2} = \frac{\frac{7}{5}}{2} = \frac{7}{5} \times \frac{1}{2} = \frac{7}{10} $$ So, \(C = \frac{7}{10}\). Among the options \(-\frac{7}{5}, -\frac{7}{10}, \frac{16}{5}\), the coordinate is \(-\frac{7}{10}\) if sign is negative or \(\frac{7}{10}\) positive. Since calculation gives positive \(\frac{7}{10}\), and user provided \(-\frac{7}{10}\), it might be a sign error. Hence, midpoint coordinate is \(\boxed{\frac{7}{10}}\). Final answers summary: 1) \(a = 2\) or \(6\) 2) \(6\) even three-digit numbers 3) 5,307 is an integer (اصم) 4) Midpoint coordinate \(\frac{7}{10}\)