'132' in base 4 is calculated as 1*4^2 + 3*4^1 + 2*4^0 = 16 + 12 + 2 = 30.

The sum of '12' and '21' in base-4 is '30' in base-4.

2*5^2 + 4*5^1 + 3*5^0 = 50 + 20 + 3 = 73.

In base-5, '34' is 3*5 + 4 = 19 and '21' is 2*5 + 1 = 11. Their sum is 30, which is '100' in base-5.

The base-5 number 243 converts to decimal as 2*5^2 + 4*5^1 + 3*5^0 = 50 + 20 + 3 = 73.

In base-5, the decimal number 10 is represented as 20.

In base-5, the decimal number 7 is represented as 12, since 1*5^1 + 2*5^0 = 5 + 2 = 7.

In a base-5 numeral system, valid digits are 0, 1, 2, 3, and 4; thus, 5 is not valid.

'12' in base-6 is 1*6 + 2 = 8. '3' is 3. Their product is 8*3 = 24, which is '42' in base-6.

7*5 = 35 in decimal, which is '43' in base-8.

The probability that both are smokers is 0.3 * 0.3 = 0.09.

The probability that it does not rain = 1 - P(rain) = 1 - 0.1 = 0.9.

Let the number of managers be x. According to the ratio, 5/2 = 70/x. Cross-multiplying gives 5x = 140, so x = 28.

Since 125 can be expressed as 5^3, we have 5^(x+1) = 5^3, thus x + 1 = 3, leading to x = 2.

The remainder when 123 is divided by 10 is 3, as 123 = 10*12 + 3.

When 37 is divided by 5, it goes 7 times (7*5=35) with a remainder of 2 (37-35=2).

The probability of losing all 5 games is (1 - 0.3)^5 = 0.168. Therefore, the probability of winning at least once is 1 - 0.168 = 0.832, which rounds to 0.836.

The relationship of direct proportionality is represented by the equation x = ky, where k is a constant.

Total parts = 5 + 3 = 8. Volume of milk = (5/8) * 32 = 20 liters.

Total parts = 5 + 3 = 8. Volume of milk = (5/8) * 64 = 40 liters.

Initial sugar = 1 liter, water = 16 liters. After adding 2 liters of sugar, the new ratio is 3:16, which simplifies to 1:5.

Let the initial amount of sugar be x liters and water be 4x liters. After adding 2 liters of sugar, the new ratio becomes (x + 2) : 4x.

Initial sugar = 1 part, water = 4 parts. Total = 5 parts. New sugar = 2 liters, water = 8 liters. Ratio = 2:8 = 1:4.

Initially, there is 1 part sugar and 4 parts water, totaling 5 parts. In 20 liters, there are 4 liters of sugar and 16 liters of water. After adding 2 liters of sugar, the new ratio is 6:16, which simplifies to 1:5.

In modular arithmetic, if a ≡ b (mod m), then (a - b) is divisible by m. Here, 7 - 3 = 4, which is divisible by 4.

'101' in binary is 5, so '111' in binary is 7.

'101' in binary is 5 in decimal. '110' in binary is 6 in decimal.

In base 5, '123' = 1*25 + 2*5 + 3 = 27.

'210' in base 3 is 2*3^2 + 1*3^1 + 0*3^0 = 18 + 3 + 0 = 21. Therefore, '102' in base 3 is 1*3^2 + 0*3^1 + 2*3^0 = 9 + 0 + 2 = 11.

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