7.4 Q-8 | Mathematical Induction And Binomial Theorem | Mathematics 11

Exercise 7.4 — Question 8

This question applies the Binomial Theorem to solve problems involving remainders, last digits, and divisibility of large powers.

Key Technique: Rewriting as a Binomial

To apply the Binomial Theorem to a large power, rewrite the base as a sum or difference involving a convenient number:

$a^{n} = (b + r)^{n}$

where $b$ is chosen so that $b$ is divisible by the divisor (or a power of it), and $r$ is small (often 1 or −1).

Expanding: $a^{n} = (0 n) b^{n} + (1 n) b^{n - 1} r + \dots + (n - 1 n) b r^{n - 1} + r^{n}$

All terms except the last ( $r^{n}$ ) contain $b$ as a factor, so they are divisible by $b$ .

Finding the Remainder

Method: Write $a = (d \cdot k + r)$ so that $d$ divides all terms except $r^{n}$ .

Example: Find the remainder when $7^{100}$ is divided by $6$ .

$7^{100} = (6 + 1)^{100} = \sum_{k = 0}^{100} (k 100) 6^{k} \cdot 1^{100 - k}$

Every term with $k \geq 1$ contains $6$ as a factor. The only term without $6$ is the $k = 0$ term:

$7^{100} = 6 M + 1 for some integer M$

$∴ Remainder = 1$

Finding the Last Digit

The last digit of $a^{n}$ equals the remainder when $a^{n}$ is divided by 10.

Example: Find the last digit of $3^{100}$ .

Note that $3^{4} = 81$ , so $3^{4} \equiv 1 (mod 10)$ .

$3^{100} = (3^{4})^{25} = 8 1^{25}$

Write $81 = 80 + 1$ :

$8 1^{25} = (80 + 1)^{25} = 80 M + 1$

$∴ Last digit of 3^{100} = 1$

Testing Divisibility

Example: Show that $1 0^{n} - 1$ is divisible by $9$ for all positive integers $n$ .

$1 0^{n} = (9 + 1)^{n} = \sum_{k = 0}^{n} (k n) 9^{k}$

All terms with $k \geq 1$ are divisible by $9$ . The $k = 0$ term is $1$ .

$1 0^{n} = 9 M + 1 ⟹ 1 0^{n} - 1 = 9 M$

$∴ 9 ∣ (1 0^{n} - 1) ✓$

Comparing Large Numbers

To compare two large numbers, expand both using the Binomial Theorem and compare the dominant terms.

Example: Which is larger: $10 0^{50}$ or $5 0^{100}$ ?

Take the ratio or use logarithms alongside binomial expansion to determine the dominant growth.

Summary of Steps

Goal	Strategy
Find remainder of $a^{n} \div d$	Write $a = d \cdot k \pm 1$ , expand, isolate constant term
Find last digit of $a^{n}$	Find remainder when $a^{n} \div 10$
Test divisibility by $d$	Write $a = d \cdot k + r$ , show $a^{n} - r^{n}$ divisible by $d$
Compare large numbers	Expand and compare leading terms

Exercise 7.4 — Question 8

This question applies the Binomial Theorem to solve problems involving remainders, last digits, and divisibility of large powers.

Key Technique: Rewriting as a Binomial

To apply the Binomial Theorem to a large power, rewrite the base as a sum or difference involving a convenient number:

$a^{n} = (b + r)^{n}$

where $b$ is chosen so that $b$ is divisible by the divisor (or a power of it), and $r$ is small (often 1 or −1).

Expanding: $a^{n} = (0 n) b^{n} + (1 n) b^{n - 1} r + \dots + (n - 1 n) b r^{n - 1} + r^{n}$

All terms except the last ( $r^{n}$ ) contain $b$ as a factor, so they are divisible by $b$ .

Finding the Remainder

Method: Write $a = (d \cdot k + r)$ so that $d$ divides all terms except $r^{n}$ .

Example: Find the remainder when $7^{100}$ is divided by $6$ .

$7^{100} = (6 + 1)^{100} = \sum_{k = 0}^{100} (k 100) 6^{k} \cdot 1^{100 - k}$

Every term with $k \geq 1$ contains $6$ as a factor. The only term without $6$ is the $k = 0$ term:

$7^{100} = 6 M + 1 for some integer M$

$∴ Remainder = 1$

Finding the Last Digit

The last digit of $a^{n}$ equals the remainder when $a^{n}$ is divided by 10.

Example: Find the last digit of $3^{100}$ .

Note that $3^{4} = 81$ , so $3^{4} \equiv 1 (mod 10)$ .

$3^{100} = (3^{4})^{25} = 8 1^{25}$

Write $81 = 80 + 1$ :

$8 1^{25} = (80 + 1)^{25} = 80 M + 1$

$∴ Last digit of 3^{100} = 1$

Testing Divisibility

Example: Show that $1 0^{n} - 1$ is divisible by $9$ for all positive integers $n$ .

$1 0^{n} = (9 + 1)^{n} = \sum_{k = 0}^{n} (k n) 9^{k}$

All terms with $k \geq 1$ are divisible by $9$ . The $k = 0$ term is $1$ .

$1 0^{n} = 9 M + 1 ⟹ 1 0^{n} - 1 = 9 M$

$∴ 9 ∣ (1 0^{n} - 1) ✓$

Comparing Large Numbers

To compare two large numbers, expand both using the Binomial Theorem and compare the dominant terms.

Example: Which is larger: $10 0^{50}$ or $5 0^{100}$ ?

Take the ratio or use logarithms alongside binomial expansion to determine the dominant growth.

Summary of Steps

Goal	Strategy
Find remainder of $a^{n} \div d$	Write $a = d \cdot k \pm 1$ , expand, isolate constant term
Find last digit of $a^{n}$	Find remainder when $a^{n} \div 10$
Test divisibility by $d$	Write $a = d \cdot k + r$ , show $a^{n} - r^{n}$ divisible by $d$
Compare large numbers	Expand and compare leading terms