CAT Rules to find the unit digit of a large number. - Practice Questions & MCQ

The concept of cyclicity in number theory is a powerful tool in determining the unit digit of a number raised to a certain power (exponent). By observing the pattern of the last digits of the power sequence, we notice that after a certain point, the unit digit repeats, creating a cycle. This cycle, known as 'cyclicity', varies for different unit digits from 0-9.

Cyclicity Pattern

1. Numbers ending in 0, 1, 5, or 6 always have cyclicity of 1. This means the unit digit remains the same no matter the exponent.

Example: . The unit digit is always 0.

2. Numbers ending in 2, 3, 7, or 8 have a cyclicity of 4. This means that the unit digit repeats every four exponents.

Example:  and so on. The unit digit repeats every four powers (2,4,8,6).

3. Numbers ending in 4 or 9 have a cyclicity of 2. This means the unit digit repeats every two exponents.

Example: , and so on. The unit digit alternates between 4 and 6.

Applying Cyclicity:

When you're asked to find the unit digit of a large exponent, you can simplify the problem by using cyclicity.

1. Identify the unit digit of the base number.

2. Determine the cyclicity pattern of the unit digit.

3. Divide the exponent by the length of the cyclicity and consider the remainder.

4. If the remainder is 0, consider the last digit in the cyclicity pattern. If the remainder is any other number, consider the corresponding position in the cyclicity pattern.

Example: Find the unit digit of 

The base number ends with 7. The cyclicity of 7 is 4 with a pattern of (7, 9, 3, 1). 

Divide 94 by 4, the remainder is 2. Thus, the unit digit of  will be the second number in the cyclicity pattern of 7 which is 9.

Key Takeaways:

- Cyclicity is a repetitive pattern of unit digits in a power sequence.

- Cyclicity patterns vary for numbers with different unit digits.

- Cyclicity is a powerful tool for simplifying calculations involving large exponents, particularly when only the unit digit or last two digits are required.

Note:

For the last two digits, the concept gets a bit more complex as it involves both the unit digit and the tens place. Typically, we need to understand modular arithmetic and Euler's theorem, which we will discuss in coming lectures.