'110' (6 in decimal) + '101' (5 in decimal) = '1011' (11 in decimal).

The expression with a coefficient of 5 for x and a constant term of -3 is 5x - 3.

Let the base be 'b'. Then, A3 = 10*b + 3 = 163. Solving gives b = 12.

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.

If the ratio of boys to girls is 3:2, then for every 3 boys, there are 2 girls. If there are 30 boys, we can set up the proportion: 3/2 = 30/x. Solving for x gives us x = 20. Therefore, there are 20 girls.

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).

Let the score in the third subject be x. The average is (90 + 80 + x) / 3 = 85. Solving gives x = 90.

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.

If A is a subset of B, it means that every element of A is contained within B.

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

Total parts = 2 + 3 = 5. Y = (3/5) * 50 = 30 liters.

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

If milk is 5 parts, then water is 3 parts. Total parts = 5 + 3 = 8. Water = (3/5) * 40 = 24 liters.

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

Initial sugar = 1 part, water = 4 parts (8 liters). After adding 2 liters of sugar, 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.

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.

Let the initial amounts be 2x and 3x. After adding 5 liters to the first component, the new ratio becomes (2x + 5):3x. Solving gives 3:2.

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

Since 7 - 3 = 4, n must be a divisor of 4. Therefore, n can be 1, 2, or 4, but for the congruence to hold, n must be greater than 4.

To convert from base 5 to decimal, calculate: 2*5^2 + 4*5^1 + 3*5^0 = 2*25 + 4*5 + 3*1 = 50 + 20 + 3 = 73.

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

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

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