The b-adic representation is a way of representing a non-negative integer using a base b. It involves expressing the integer as a sum of multiples of powers of b. For example, in base 10, the number 365 can be represented as 3*10^2 + 6*10^1 + 5*10^0. This representation allows for easy conversion between different bases and is commonly used in computer science and number theory. **

To multiply 7-adic numbers, you can use the same method as multiplying regular integers, but with carrying out the operations modulo 7 at each step. Start by multiplying the corresponding digits of the two numbers, then carry over any excess to the next digit. Repeat this process for each digit position. Remember that in the 7-adic number system, each digit can only be between 0 and 6. **

A b-adic fraction is a way of representing a real number using a base b expansion. It is similar to a decimal expansion, but instead of using base 10, it uses base b. The b-adic expansion of a number consists of a sequence of digits, each of which represents a multiple of a power of b. The expansion can be finite or infinite, and it can be used to represent both rational and irrational numbers. **

To convert recurring fractions into q-adic representation, we can use the formula for converting a recurring fraction to a q-adic number. First, we express the recurring fraction as a continued fraction. Then, we use the formula for converting a continued fraction to a q-adic number, which involves finding the convergents of the continued fraction and expressing them in terms of powers of q. This process allows us to represent the recurring fraction as a q-adic number. **

The popularity of cycling or skiing can vary depending on the location and time of year. In general, cycling tends to be more popular in urban areas and during the warmer months, while skiing is more popular in mountainous regions and during the winter. However, it's difficult to determine which activity is followed more overall without specific data on participation rates in different regions and seasons. **

The excitement level of ice hockey versus soccer is subjective and can vary depending on personal preferences. Ice hockey is known for its fast-paced, physical nature with constant action and scoring opportunities. On the other hand, soccer is a more strategic game with longer periods of build-up play and intense moments during goal-scoring opportunities. Both sports have passionate fan bases and offer unique thrills, making it difficult to definitively say which is more exciting. **

It is highly unlikely that soccer will replace American football, basketball, baseball, and ice hockey as the most popular sports in the USA. These sports have deep-rooted cultural and historical significance in American society, and they have large and dedicated fan bases. Additionally, American football, basketball, baseball, and ice hockey have well-established professional leagues and infrastructure in the USA, which would make it difficult for soccer to overtake them in popularity. While soccer has been growing in popularity in the USA, it is more likely to continue to coexist alongside these other sports rather than replace them. **

Ice hockey is often perceived as more brutal than soccer due to the physical nature of the sport, which includes body checking and fighting. However, both sports have their own unique physical demands and risks of injury. While ice hockey may have more violent collisions, soccer players also face the risk of serious injuries such as concussions and ACL tears. Ultimately, the level of brutality in each sport can vary depending on the intensity of the competition and the actions of the players involved. **