Lxample: Prime Factorization of 120
In each case find the HCF and LCM of the numbers
780 and 7700
380 and 1360
Using prime factors to simplity fractions
a. Write the fraction in its simplest form, without the use of a calculator

Question

Lxample: Prime Factorization of 120
In each case find the HCF and LCM of the numbers
780 and 7700
380 and 1360
Using prime factors to simplity fractions
a. Write the fraction

in its simplest form, without the use of a calculator

UpStudy AI · Accepted Answer

Sure, let's work through each of the problems step by step using prime factorization.

---

### 1. **Finding the HCF and LCM of 780 and 7700**

**Prime Factorization:**

- **Prime Factors of 780:**
  $$
  780 = 2 \times 390 \
  390 = 2 \times 195 \
  195 = 3 \times 65 \
  65 = 5 \times 13 \
  $$
  So, the prime factorization of **780** is:
  $$
  780 = 2^2 \times 3 \times 5 \times 13
  $$

- **Prime Factors of 7700:**
  $$
  7700 = 2 \times 3850 \
  3850 = 2 \times 1925 \
  1925 = 5 \times 385 \
  385 = 5 \times 77 \
  77 = 7 \times 11 \
  $$
  So, the prime factorization of **7700** is:
  $$
  7700 = 2^2 \times 5^2 \times 7 \times 11
  $$

**Highest Common Factor (HCF):**

The HCF is the product of the smallest powers of all common prime factors.

- Common prime factors of 780 and 7700:
  - $$2^2$$
  - $$5$$

Thus,
$$
\text{HCF} = 2^2 \times 5 = 4 \times 5 = 20
$$

**Least Common Multiple (LCM):**

The LCM is the product of the highest powers of all prime factors present in either number.

- Prime factors to consider:
  - $$2^2$$ (from both)
  - $$3$$ (from 780)
  - $$5^2$$ (from 7700)
  - $$7$$ (from 7700)
  - $$11$$ (from 7700)
  - $$13$$ (from 780)

Thus,
$$
\text{LCM} = 2^2 \times 3 \times 5^2 \times 7 \times 11 \times 13
$$

Calculating step-by-step:
$$
2^2 = 4 \
5^2 = 25 \
4 \times 3 = 12 \
12 \times 25 = 300 \
300 \times 7 = 2100 \
2100 \times 11 = 23100 \
23100 \times 13 = 300300 \
$$
So,
$$
\text{LCM} = 300{,}300
$$

---

### 2. **Finding the HCF and LCM of 380 and 1360**

**Prime Factorization:**

- **Prime Factors of 380:**
  $$
  380 = 2 \times 190 \
  190 = 2 \times 95 \
  95 = 5 \times 19 \
  $$
  So, the prime factorization of **380** is:
  $$
  380 = 2^2 \times 5 \times 19
  $$

- **Prime Factors of 1360:**
  $$
  1360 = 2 \times 680 \
  680 = 2 \times 340 \
  340 = 2 \times 170 \
  170 = 2 \times 85 \
  85 = 5 \times 17 \
  $$
  So, the prime factorization of **1360** is:
  $$
  1360 = 2^4 \times 5 \times 17
  $$

**Highest Common Factor (HCF):**

- Common prime factors of 380 and 1360:
  - $$2^2$$
  - $$5$$

Thus,
$$
\text{HCF} = 2^2 \times 5 = 4 \times 5 = 20
$$

**Least Common Multiple (LCM):**

- Prime factors to consider:
  - $$2^4$$ (from 1360)
  - $$5$$ (from both)
  - $$17$$ (from 1360)
  - $$19$$ (from 380)

Thus,
$$
\text{LCM} = 2^4 \times 5 \times 17 \times 19
$$

Calculating step-by-step:
$$
2^4 = 16 \
16 \times 5 = 80 \
80 \times 17 = 1360 \
1360 \times 19 = 25{,}840 \
$$
So,
$$
\text{LCM} = 25{,}840
$$

---

### 3. **Simplifying the Fraction $$ \frac{330}{380} $$ Using Prime Factors**

**Prime Factorization:**

- **Numerator (330):**
  $$
  330 = 2 \times 165 \
  165 = 3 \times 55 \
  55 = 5 \times 11 \
  $$
  So,
  $$
  330 = 2 \times 3 \times 5 \times 11
  $$

- **Denominator (380):**
  As previously determined,
  $$
  380 = 2^2 \times 5 \times 19
  $$

**Simplifying the Fraction:**

1. **Identify Common Prime Factors:**
   - Both numerator and denominator have the prime factors $$2$$ and $$5$$.

2. **Determine the Greatest Common Divisor (GCD):**
   - From the prime factors:
     - $$2$$ (minimum exponent is $$1$$)
     - $$5$$ (minimum exponent is $$1$$)
   - So, GCD $$= 2 \times 5 = 10$$

3. **Divide Both Numerator and Denominator by the GCD:**
   $$
   \frac{330}{380} = \frac{330 \div 10}{380 \div 10} = \frac{33}{38}
   $$

4. **Final Simplified Form:**
   $$
   \frac{330}{380} = \frac{33}{38}
   $$
   
   Since $$33$$ and $$38$$ share no common prime factors (other than $$1$$), the fraction is in its simplest form.

---

**Summary:**

- **HCF of 780 and 7700:** 20
- **LCM of 780 and 7700:** 300,300
- **HCF of 380 and 1360:** 20
- **LCM of 380 and 1360:** 25,840
- **Simplified Fraction:** $$ \frac{330}{380} = \frac{33}{38} $$
- **HCF of 780 and 7700:** 20 - **LCM of 780 and 7700:** 300,300 - **HCF of 380 and 1360:** 20 - **LCM of 380 and 1360:** 25,840 - **Simplified Fraction:** $$ \frac{330}{380} = \frac{33}{38} $$

Lxample: Prime Factorization of 120
In each case find the HCF and LCM of the numbers
780 and 7700
380 and 1360
Using prime factors to simplity fractions
a. Write the fraction in its simplest form, without the use of a calculator

Upstudy AI Solution

Answer

Solution

Mind Expander

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Latest Arithmetic Questions

Lxample: Prime Factorization of 120 In each case find the HCF and LCM of the numbers 780 and 7700 380 and 1360 Using prime factors to simplity fractions a. Write the fraction in its simplest form, without the use of a calculator

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Arithmetic Questions

Lxample: Prime Factorization of 120
In each case find the HCF and LCM of the numbers
780 and 7700
380 and 1360
Using prime factors to simplity fractions
a. Write the fraction in its simplest form, without the use of a calculator