# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the application of the Binomial Theorem and Mathematical Induction to solve problems in number theory. - **Cyclicity of Unit Digits:** Understanding that the unit digits of powers of a number repeat in a cycle. This is used to find the last digit of a number raised to a large power. - **Remainder Theorem using Binomial Expansion:** A key technique is to express a number *A* as a multiple of the divisor *d* plus or minus a remainder, i.e., $A = (kd \pm r)$. The binomial expansion of $A^n = (kd \pm r)^n$ can then be used to find the remainder when $A^n$ is divided by *d*. - **Finding Last Two Digits:** This is an extension of the remainder concept, where the goal is to find the remainder when a number is divided by 100. Expressing the base as a sum or difference from a multiple of 10 (e.g., $29^{10} = (30-1)^{10}$) simplifies the calculation. - **Divisibility Proofs:** Using binomial expansion or mathematical induction to prove that a given expression is divisible by a specific integer for all positive integers *n*. - **Comparing Magnitudes:** Applying binomial expansion to compare the size of two numbers, especially when one involves a large exponent, like comparing $(1.1)^{10000}$ with 1000. --- # **Important Formulas** Below are the key formulas and principles used in this exercise: | Description | Formula / Principle | | :--- | :--- | | Binomial Theorem | $(a+b)^{n} = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \ldots + \binom{n}{n}b^n$ | | Remainder Principle | When $(kd+r)^n$ is divided by *d*, the remainder is the same as the remainder when $r^n$ is divided by *d*. | | Divisibility of Sums of Powers (Odd Exponent) | $a^n + b^n$ is divisible by $a+b$ if *n* is an odd integer. | | Divisibility of Differences of Powers | $a^n - b^n$ is always divisible by $a-b$ for any positive integer *n*. | | Principle of Mathematical Induction | To prove a statement P(n): 1. **Basis Step:** Show P(1) is true. 2. **Inductive Step:** Assume P(k) is true and use this to prove that P(k+1) is also true. | --- # **Summary** This exercise is a practical application of the Binomial Theorem and Mathematical Induction to solve a variety of number theory problems. The questions challenge the user to move beyond simple expansions and apply these tools to find remainders, determine unit digits, prove divisibility, and compare the magnitude of large numbers. A central strategy involves cleverly rewriting numbers to fit the $(a+b)^n$ form, where 'a' is a multiple of the divisor. This simplifies complex calculations by showing that most terms in the expansion are divisible, leaving only the last one or two terms to determine the remainder or final digits. The inclusion of proofs by induction reinforces logical reasoning and demonstrates the versatility of these mathematical principles. --- # **Reference Videos**