Exercise 7.3 — Question 9

Topic: Applications of the Binomial Theorem

This question covers three key applications of the Binomial Theorem:

Expanding and simplifying binomial expressions
Finding remainders when large powers are divided by a number
Finding the last digit (units digit) of large powers and testing divisibility

Key Formula

The Binomial Theorem states:

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

where $(r n) = \frac{n !}{r ! ( n - r )!}$ are the binomial coefficients.

Application 1: Expansion and Simplification (SLO M-11-A-37)

To expand $(a + b)^{n}$ , apply the Binomial Theorem term by term. The general term is:

$T_{r + 1} = (r n) a^{n - r} b^{r}$

Example: Expand $(x + 2)^{4}$

$(x + 2)^{4} = (0 4) x^{4} + (1 4) x^{3} (2) + (2 4) x^{2} (4) + (3 4) x (8) + (4 4) (16)$ $= x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16$

Application 2: Finding Remainders (SLO M-11-A-39)

Strategy: Rewrite the base as $(divisor \pm 1)$ or a convenient form, then expand using the Binomial Theorem. All terms except the last will be divisible by the divisor.

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

Write $7 = 6 + 1$ , so: $7^{100} = (6 + 1)^{100} = \sum_{r = 0}^{100} (r 100) 6^{r} \cdot 1^{100 - r}$ $= (0 100) (1) + (1 100) (6) + (2 100) (6^{2}) + \dots + 6^{100}$

Every term except the first contains a factor of $6$ , so: $7^{100} = 1 + 6 k for some integer k$

$Remainder = 1$

Application 3: Last Digit and Divisibility (SLO M-11-A-40)

Strategy for last digit: Rewrite the base as $(multiple of 10 \pm d)$ where $d$ is a small number. After expansion, only the last term (which doesn't contain 10) determines the units digit.

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

Note that $3^{4} = 81$ , so $3^{100} = (3^{4})^{25} = 8 1^{25}$ .

Write $81 = 80 + 1$ : $8 1^{25} = (80 + 1)^{25} = \sum_{r = 0}^{25} (r 25) 8 0^{r}$ $= 1 + 25 (80) + (2 25) (80)^{2} + \dots$

All terms except $1$ are multiples of $80$ (hence multiples of $10$ ), so the units digit is: $1$

Divisibility Test Example: Show that $5^{2 n} - 1$ is divisible by $24$ .

Write $5^{2 n} = 2 5^{n} = (24 + 1)^{n}$ : $(24 + 1)^{n} = 1 + n (24) + (2 n) (24)^{2} + \dots$ $5^{2 n} - 1 = 24 n + (2 n) (24)^{2} + \dots = 24 [n + (2 n) (24) + \dots]$

Hence $5^{2 n} - 1$ is divisible by $24$ . $✓$

Summary Table

Application	Key Trick	Result
Remainder	Write base as (divisor ± 1), expand	Constant term gives remainder
Last digit	Write base as (multiple of 10 ± d), expand	Constant term gives units digit
Divisibility	Write base as (divisor + 1), expand	Factor out divisor from all non-constant terms

Application 2: Finding Remainders (SLO M-11-A-39)

Strategy: Rewrite the base as

(divisor \pm 1)

or a convenient form, then expand using the Binomial Theorem. All terms except the last will be divisible by the divisor.

Example: Find the remainder when

7^{100}

is divided by

6

Write

7 = 6 + 1

, so:

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

= (0 100) (1) + (1 100) (6) + (2 100) (6^{2}) + \dots + 6^{100}

Every term except the first contains a factor of

6

, so:

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

Remainder = 1

Application 3: Last Digit and Divisibility (SLO M-11-A-40)

Strategy for last digit: Rewrite the base as

(multiple of 10 \pm d)

where

d

is a small number. After expansion, only the last term (which doesn't contain 10) determines the units digit.

Example: Find the last digit of

3^{100}

Note that

3^{4} = 81

, so

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

Write

81 = 80 + 1

8 1^{25} = (80 + 1)^{25} = \sum_{r = 0}^{25} (r 25) 8 0^{r}

= 1 + 25 (80) + (2 25) (80)^{2} + \dots

All terms except

1

are multiples of

80

(hence multiples of

10

), so the units digit is:

1

Divisibility Test Example: Show that

5^{2 n} - 1

is divisible by

24

Write

5^{2 n} = 2 5^{n} = (24 + 1)^{n}

(24 + 1)^{n} = 1 + n (24) + (2 n) (24)^{2} + \dots

5^{2 n} - 1 = 24 n + (2 n) (24)^{2} + \dots = 24 [n + (2 n) (24) + \dots]

Hence

5^{2 n} - 1

is divisible by

24

✓

Application

Key Trick

Result

Remainder

Write base as (divisor ± 1), expand

Constant term gives remainder

Last digit

Write base as (multiple of 10 ± d), expand

Constant term gives units digit

Divisibility

Write base as (divisor + 1), expand

Factor out divisor from all non-constant terms

7.3 Q-9

Exercise 7.3 — Question 9

Topic: Applications of the Binomial Theorem

Key Formula

Application 1: Expansion and Simplification (SLO M-11-A-37)

Application 2: Finding Remainders (SLO M-11-A-39)

Application 3: Last Digit and Divisibility (SLO M-11-A-40)

Summary Table

7.3 Q-9

Exercise 7.3 — Question 9

Topic: Applications of the Binomial Theorem

Key Formula

Application 1: Expansion and Simplification (SLO M-11-A-37)

Application 2: Finding Remainders (SLO M-11-A-39)

Application 3: Last Digit and Divisibility (SLO M-11-A-40)

Summary Table