https://file.aiquickdraw.com/mathbot/file/1726662026984jtrp3spj.jpg

Let's break down each problem from the image.

### E2. Express each of the following as a product of prime factors only in exponential form:

**i) 108 × 192**

We'll factorize each number:
- \(108 = 2^2 \times 3^3\)
- \(192 = 2^6 \times 3\)

Now multiply the factorizations:
\[
108 \times 192 = (2^2 \times 3^3) \times (2^6 \times 3) = 2^{2+6} \times 3^{3+1} = 2^8 \times 3^4
\]

**ii) 270**

Factorize 270:
\[
270 = 2 \times 3^3 \times 5
\]

**iii) 729 × 64**

Factorize each number:
- \(729 = 3^6\)
- \(64 = 2^6\)

Now multiply the factorizations:
\[
729 \times 64 = 3^6 \times 2^6
\]

### E3. Find the unit digit of:

**a) \(3^4\)**

\[
3^4 = 81 \quad \text{(unit digit is 1)}
\]

**b) \(3^{2025}\)**

The powers of 3 repeat in cycles of 4 for the unit digit: \(3, 9, 7, 1\).
To find the unit digit of \(3^{2025}\), divide 2025 by 4 and find the remainder:
\[
2025 \div 4 = 506 \, \text{remainder} \, 1
\]
So, the unit digit is the same as \(3^1\), which is **3**.

**c) \(2^4\)**

\[
2^4 = 16 \quad \text{(unit digit is 6)}
\]

**d) \(2^{2024}\)**

The powers of 2 repeat in cycles of 4 for the unit digit: \(2, 4, 8, 6\).
To find the unit digit of \(2^{2024}\), divide 2024 by 4 and find the remainder:
\[
2024 \div 4 = 506 \, \text{remainder} \, 0
\]
So, the unit digit is the same as \(2^4\), which is **6**.

### E4. Find the remainder for:

**a) \(133^2\) divided by 12**

First, calculate \(133 \mod 12\):
\[
133 \div 12 = 11 \quad \text{(remainder is 1)}
\]
Now, square the remainder:
\[
133^2 \mod 12 = 1^2 \mod 12 = 1
\]

**b) \(133^{2024}\) divided by 12**

Since \(133 \mod 12 = 1\), we have:
\[
133^{2024} \mod 12 = 1^{2024} \mod 12 = 1
\]

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Let me know if you'd like further details, and feel free to ask any questions!

Here are 5 related questions for practice:

1. What is the prime factorization of 540 in exponential form?
2. Find the unit digit of \(4^{2025}\).
3. Factorize \(360\) into prime factors in exponential form.
4. Find the remainder when \(144^2\) is divided by 9.
5. Determine the remainder when \(5^{50}\) is divided by 12.

**Tip:** For powers involving units digits, always check for cyclic patterns!

E2. Express each of the following as a product of prime factors only in exponential form. E3. Find the unit digit of 3^4, 3^2025, 2^4, 2^2024. E4. Find the remainder for 133^2 divided by 12 and 133^2024 divided by 12.

Solve problems involving prime factorization, finding unit digits of powers, and modular arithmetic. Learn how to express numbers in exponential form, find unit digits for powers like 3^2025, and calculate remainders such as 133^2 divided by 12.

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