'B' represents 11 in the same system where 'A' is 10.

'B' (18) is a multiple of 'A' (12) since 18 can be expressed as 12 multiplied by 1.5.

'A' can be 1, 2, 3, 4, 6, or 12 (factors of 12) and 'B' can be 3, 6, 9, 12, etc. The only combination that fits is A=3 and B=12, which gives us 36.

In this case, A must be a factor of 12 (1, 2, 3, 4, 6, 12) and B must be a multiple of 3 (3, 6, 9, 12). The pair (2, 6) satisfies both conditions.

'A' can be 3 or 4 (factors of 12), and 'B' can be 3, 6, or 9 (multiples of 3). The only combination that fits is 3 and 4, which gives us 24.

The number 1234 is expressed in base 8, as indicated by the powers of 8 used in its representation.

If the original number is x, then x + 12 is a multiple of 5. This implies that x must be one more than a multiple of 5.

If we add 12 to 8, we get 20, which is a multiple of 5.

A number is divisible by 3 if the sum of its digits is divisible by 3.

'B2' in this system is 11*12^1 + 2*12^0 = 132 + 2 = 134 in decimal.

'210' in base 4 is calculated as 2*4^2 + 1*4^1 + 0*4^0 = 32 + 4 + 0 = 36.

'123' in base 3 = 1*3^2 + 2*3^1 + 3*3^0 = 27. '321' in base 3 = 3*3^2 + 2*3^1 + 1*3^0 = 81.

'23' in this system is 15. '32' would represent 22 in decimal.

Initially, there are 36 boys and 24 girls. After the changes, there will be 46 boys and 19 girls. The new ratio is 46:19, which simplifies to approximately 4:3.

Initially, there are 36 boys and 24 girls. After increasing boys by 5, there will be 41 boys and 24 girls, giving a new ratio of 41:24, which simplifies to approximately 4:3.

Initially, there are 36 boys and 24 girls. After increasing the boys by 10, there will be 46 boys and 24 girls, giving a new ratio of 46:24, which simplifies to 4:3.

Let the number of boys be 3x and girls be 2x. Then, 3x + 2x = 50, which gives x = 10. Therefore, boys = 3x = 30.

If there are 10 students with 2 pencils, then there are 10 students with 3 pencils, totaling 30 pencils.

The least common multiple of 4 and 6 is 12. Therefore, any multiple of 12 is valid, and 24 is a multiple of 12.

The multiples of 3 that can divide 24 are 3, 6, 9, and 12. The maximum number of students that can receive pencils is 8 (3 pencils each).

The maximum number of students is 12, as 48 ÷ 4 = 12.

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. 10 is not a factor of 48.

5 is not a factor of 36, hence it cannot be a possible number of chairs per row.

The number 10 is not a factor of 36, hence it cannot be a group size.

5 rows of 5 students would require 25 students, which is not possible with only 24 students.

In modular arithmetic with a modulus of 12 (for hours), 9 + 15 = 24, and 24 mod 12 = 0, which corresponds to 12 o'clock.

In a 12-hour clock, 3 + 10 = 13, and 13 mod 12 = 1.

15 hours from 9 o'clock is calculated as (9 + 15) mod 12 = 24 mod 12 = 0, which is 12 o'clock.

'1A' in hexadecimal is 1*16 + 10 = 26 in decimal.

In hexadecimal, 'A' represents 10 and '3' represents 3. Therefore, A3 = (10 * 16^1) + (3 * 16^0) = 160 + 3 = 163.