Exercise 7.4 — Question 2

This exercise focuses on advanced applications of the Binomial Theorem for:

Finding remainders when large powers are divided by a number
Finding the last (units) digit of a large power
Testing divisibility of large numbers
Comparing two large numbers

Key Technique: Rewriting as a Binomial

The core strategy is to express the base as $(1 + a)^{n}$ or $(d \cdot m + r)^{n}$ so that the Binomial expansion isolates the remainder or last digit.

$(a + b)^{n} = \sum_{k = 0}^{n} (k n) a^{n - k} b^{k}$

1. Finding the Remainder

Method: Write the base as $(d + r)$ where $d$ is the divisor (or a multiple of it).

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

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

All terms with $k \geq 1$ contain $6$ as a factor, so they are divisible by $6$ . The only term not divisible by $6$ is the $k = 0$ term:

$(0 100) \cdot 6^{0} = 1$

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

Remainder $= 1$

2. Finding the Last (Units) Digit

Method: Write the base as $(10 + r)^{n}$ or identify the units digit pattern.

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

$3^{100} = (3^{4})^{25} = 8 1^{25} = (80 + 1)^{25}$

Expanding: $(80 + 1)^{25} = \sum_{k = 0}^{25} (k 25) 8 0^{k} \cdot 1^{25 - k}$

All terms with $k \geq 1$ contain $80$ (hence $10$ ) as a factor, so their units digit is $0$ . The $k = 0$ term is $1$ .

$∴ Last digit of 3^{100} = 1$

3. 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 $1$ .

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

$∴ 9 ∣ (1 0^{n} - 1) ✓$

4. Comparing Two Large Numbers

Example: Which is greater: $2^{30}$ or $3^{20}$ ?

Rewrite both with the same exponent: $2^{30} = (2^{3})^{10} = 8^{10}, 3^{20} = (3^{2})^{10} = 9^{10}$

Since $9 > 8$ : $9^{10} > 8^{10} ⟹ 3^{20} > 2^{30}$

Summary Table

Application	Strategy
Remainder	Write base as $(d + r)^{n}$ ; all terms except last divisible by $d$
Last digit	Write base as $(10 + r)^{n}$ ; only constant term gives units digit
Divisibility	Show $n^{k} = d M + 0$ using Binomial expansion
Comparison	Rewrite both numbers with equal exponents, compare bases

1. Finding the Remainder

Method: Write the base as

(d + r)

where

d

is the divisor (or a multiple of it).

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}

All terms with

k \geq 1

contain

6

as a factor, so they are divisible by

6

. The only term not divisible by

6

is the

k = 0

term:

(0 100) \cdot 6^{0} = 1

∴ 7^{100} = 6 M + 1 (for some integer M)

Remainder

= 1

2. Finding the Last (Units) Digit

Method: Write the base as

(10 + r)^{n}

or identify the units digit pattern.

Example: Find the last digit of

3^{100}

3^{100} = (3^{4})^{25} = 8 1^{25} = (80 + 1)^{25}

Expanding:

(80 + 1)^{25} = \sum_{k = 0}^{25} (k 25) 8 0^{k} \cdot 1^{25 - k}

All terms with

k \geq 1

contain

80

(hence

10

) as a factor, so their units digit is

0

. The

k = 0

term is

1

∴ Last digit of 3^{100} = 1

Application

Strategy

Remainder

Write base as

(d + r)^{n}

; all terms except last divisible by

d

Last digit

Write base as

(10 + r)^{n}

; only constant term gives units digit

Divisibility

Show

n^{k} = d M + 0

using Binomial expansion

Comparison

Rewrite both numbers with equal exponents, compare bases

7.4 Q-2

Exercise 7.4 — Question 2

Key Technique: Rewriting as a Binomial

1. Finding the Remainder

2. Finding the Last (Units) Digit

3. Testing Divisibility

4. Comparing Two Large Numbers

Summary Table

7.4 Q-2

Exercise 7.4 — Question 2

Key Technique: Rewriting as a Binomial

1. Finding the Remainder

2. Finding the Last (Units) Digit

3. Testing Divisibility

4. Comparing Two Large Numbers

Summary Table