Question 2

We will find the prime factors of each number. Then we will use these prime factors to find all factors. Finally, we will pick out the common factors and the HCF.

Step 1 — Prime Factorization of 50 and 60

Let us break down 50 into its prime factors. We divide 50 by the smallest prime number, 2. $50 \div 2 = 25$ Now we divide 25 by the smallest prime number that goes into it, which is 5. $25 \div 5 = 5$ Finally, we divide 5 by 5. $5 \div 5 = 1$ So, the prime factors of 50 are 2, 5, and 5.

$\boxed{50 = 2 \times 5 \times 5}$

Next, let us break down 60 into its prime factors. We divide 60 by 2. $60 \div 2 = 30$ We divide 30 by 2 again. $30 \div 2 = 15$ We divide 15 by the next prime number, 3. $15 \div 3 = 5$ Finally, we divide 5 by 5. $5 \div 5 = 1$ So, the prime factors of 60 are 2, 2, 3, and 5.

$\boxed{60 = 2 \times 2 \times 3 \times 5}$

Step 2 — Listing All Factors of 50 and 60

To find all factors of 50, we use its prime factors: 2, 5, 5. The factors are 1, 2, 5, $2 \times 5 = 10$, $5 \times 5 = 25$, and $2 \times 5 \times 5 = 50$.

$\boxed{\text{Factors of } 50: 1, 2, 5, 10, 25, 50}$

To find all factors of 60, we use its prime factors: 2, 2, 3, 5. The factors are 1, 2, 3, 4 ($2 \times 2$), 5, 6 ($2 \times 3$), 10 ($2 \times 5$), 12 ($2 \times 2 \times 3$), 15 ($3 \times 5$), 20 ($2 \times 2 \times 5$), 30 ($2 \times 3 \times 5$), and 60 ($2 \times 2 \times 3 \times 5$).

$\boxed{\text{Factors of } 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}$

Step 3 — Finding Common Factors and HCF for 50 and 60

Now we look for numbers that appear in both lists of factors. The common factors are 1, 2, 5, and 10.

$\boxed{\text{Common factors of 50 and 60 are } 1, 2, 5, 10.}$ The HCF is the Highest Common Factor. It is the largest number among the common factors. From 1, 2, 5, 10, the largest is 10. We can also find HCF by multiplying common prime factors. The prime factors of 50 are $2 \times 5 \times 5$. The prime factors of 60 are $2 \times 2 \times 3 \times 5$. The common prime factors are one 2 and one 5. $HCF = 2 \times 5$ $= 10$ $\boxed{\text{HCF}(50, 60) = 10}$

Step 4 — Prime Factorization of 140 and 275

Let us break down 140 into its prime factors. $140 \div 2 = 70$ $70 \div 2 = 35$ $35 \div 5 = 7$ $7 \div 7 = 1$ So, the prime factors of 140 are 2, 2, 5, and 7.

$\boxed{140 = 2 \times 2 \times 5 \times 7}$

Next, let us break down 275 into its prime factors. 275 ends in 5, so we divide by 5. $275 \div 5 = 55$ We divide 55 by 5 again. $55 \div 5 = 11$ 11 is a prime number, so we divide by 11. $11 \div 11 = 1$ So, the prime factors of 275 are 5, 5, and 11.

$\boxed{275 = 5 \times 5 \times 11}$

Step 5 — Listing All Factors of 140 and 275

To find all factors of 140, we use its prime factors: 2, 2, 5, 7. The factors are 1, 2, 4 ($2 \times 2$), 5, 7, 10 ($2 \times 5$), 14 ($2 \times 7$), 20 ($2 \times 2 \times 5$), 28 ($2 \times 2 \times 7$), 35 ($5 \times 7$), 70 ($2 \times 5 \times 7$), and 140 ($2 \times 2 \times 5 \times 7$).

$\boxed{\text{Factors of } 140: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140}$

To find all factors of 275, we use its prime factors: 5, 5, 11. The factors are 1, 5, 11, 25 ($5 \times 5$), 55 ($5 \times 11$), and 275 ($5 \times 5 \times 11$).

$\boxed{\text{Factors of } 275: 1, 5, 11, 25, 55, 275}$

Step 6 — Finding Common Factors and HCF for 140 and 275

We look for numbers that appear in both lists of factors. The common factors are 1 and 5.

$\boxed{\text{Common factors of 140 and 275 are } 1, 5.}$ The HCF is the largest number among the common factors. From 1, 5, the largest is 5. We can also find HCF by multiplying common prime factors. The prime factors of 140 are $2 \times 2 \times 5 \times 7$. The prime factors of 275 are $5 \times 5 \times 11$. The only common prime factor is one 5. $HCF = 5$ $\boxed{\text{HCF}(140, 275) = 5}$

Step 7 — Prime Factorization of 77 and 725

Let us break down 77 into its prime factors. We divide 77 by 7. $77 \div 7 = 11$ 11 is a prime number, so we divide by 11. $11 \div 11 = 1$ So, the prime factors of 77 are 7 and 11.

$\boxed{77 = 7 \times 11}$

Next, let us break down 725 into its prime factors. 725 ends in 5, so we divide by 5. $725 \div 5 = 145$ We divide 145 by 5 again. $145 \div 5 = 29$ 29 is a prime number, so we divide by 29. $29 \div 29 = 1$ So, the prime factors of 725 are 5, 5, and 29.

$\boxed{725 = 5 \times 5 \times 29}$

Step 8 — Listing All Factors of 77 and 725

To find all factors of 77, we use its prime factors: 7, 11. The factors are 1, 7, 11, and 77 ($7 \times 11$).

$\boxed{\text{Factors of } 77: 1, 7, 11, 77}$

To find all factors of 725, we use its prime factors: 5, 5, 29. The factors are 1, 5, 25 ($5 \times 5$), 29, 145 ($5 \times 29$), and 725 ($5 \times 5 \times 29$).

$\boxed{\text{Factors of } 725: 1, 5, 25, 29, 145, 725}$

Step 9 — Finding Common Factors and HCF for 77 and 725

We look for numbers that appear in both lists of factors. Only 1 appears in both lists.

$\boxed{\text{Common factor of 77 and 725 is } 1.}$ The HCF is the largest number among the common factors. Since only 1 is common, the HCF is 1. We can also find HCF by multiplying common prime factors. The prime factors of 77 are $7 \times 11$. The prime factors of 725 are $5 \times 5 \times 29$. There are no common prime factors. When there are no common prime factors, the HCF is always 1. $HCF = 1$ $\boxed{\text{HCF}(77, 725) = 1}$

Step 10 — Prime Factorization of 370 and 592

Let us break down 370 into its prime factors. $370 \div 2 = 185$ 185 ends in 5, so we divide by 5. $185 \div 5 = 37$ 37 is a prime number, so we divide by 37. $37 \div 37 = 1$ So, the prime factors of 370 are 2, 5, and 37.

$\boxed{370 = 2 \times 5 \times 37}$

Next, let us break down 592 into its prime factors. $592 \div 2 = 296$ $296 \div 2 = 148$ $148 \div 2 = 74$ $74 \div 2 = 37$ 37 is a prime number, so we divide by 37. $37 \div 37 = 1$ So, the prime factors of 592 are 2, 2, 2, 2, and 37.

$\boxed{592 = 2 \times 2 \times 2 \times 2 \times 37}$

Step 11 — Listing All Factors of 370 and 592

To find all factors of 370, we use its prime factors: 2, 5, 37. The factors are 1, 2, 5, 10 ($2 \times 5$), 37, 74 ($2 \times 37$), 185 ($5 \times 37$), and 370 ($2 \times 5 \times 37$).

$\boxed{\text{Factors of } 370: 1, 2, 5, 10, 37, 74, 185, 370}$

To find all factors of 592, we use its prime factors: 2, 2, 2, 2, 37. The factors are 1, 2, 4 ($2 \times 2$), 8 ($2 \times 2 \times 2$), 16 ($2 \times 2 \times 2 \times 2$), 37, 74 ($2 \times 37$), 148 ($2 \times 2 \times 37$), 296 ($2 \times 2 \times 2 \times 37$), and 592 ($2 \times 2 \times 2 \times 2 \times 37$).

$\boxed{\text{Factors of } 592: 1, 2, 4, 8, 16, 37, 74, 148, 296, 592}$

Step 12 — Finding Common Factors and HCF for 370 and 592

We look for numbers that appear in both lists of factors. The common factors are 1, 2, 37, and 74.

$\boxed{\text{Common factors of 370 and 592 are } 1, 2, 37, 74.}$ The HCF is the largest number among the common factors. From 1, 2, 37, 74, the largest is 74. We can also find HCF by multiplying common prime factors. The prime factors of 370 are $2 \times 5 \times 37$. The prime factors of 592 are $2 \times 2 \times 2 \times 2 \times 37$. The common prime factors are one 2 and one 37. $HCF = 2 \times 37$ $= 74$ $\boxed{\text{HCF}(370, 592) = 74}$

Step 13 — Prime Factorization of 81 and 243

Let us break down 81 into its prime factors. We divide 81 by 3. $81 \div 3 = 27$ We divide 27 by 3. $27 \div 3 = 9$ We divide 9 by 3. $9 \div 3 = 3$ We divide 3 by 3. $3 \div 3 = 1$ So, the prime factors of 81 are 3, 3, 3, and 3.

$\boxed{81 = 3 \times 3 \times 3 \times 3}$

Next, let us break down 243 into its prime factors. We divide 243 by 3. $243 \div 3 = 81$ We divide 81 by 3. $81 \div 3 = 27$ We divide 27 by 3. $27 \div 3 = 9$ We divide 9 by 3. $9 \div 3 = 3$ We divide 3 by 3. $3 \div 3 = 1$ So, the prime factors of 243 are 3, 3, 3, 3, and 3.

$\boxed{243 = 3 \times 3 \times 3 \times 3 \times 3}$

Step 14 — Listing All Factors of 81 and 243

To find all factors of 81, we use its prime factors: 3, 3, 3, 3. The factors are 1, 3, 9 ($3 \times 3$), 27 ($3 \times 3 \times 3$), and 81 ($3 \times 3 \times 3 \times 3$).

$\boxed{\text{Factors of } 81: 1, 3, 9, 27, 81}$

To find all factors of 243, we use its prime factors: 3, 3, 3, 3, 3. The factors are 1, 3, 9 ($3 \times 3$), 27 ($3 \times 3 \times 3$), 81 ($3 \times 3 \times 3 \times 3$), and 243 ($3 \times 3 \times 3 \times 3 \times 3$).

$\boxed{\text{Factors of } 243: 1, 3, 9, 27, 81, 243}$

Step 15 — Finding Common Factors and HCF for 81 and 243

We look for numbers that appear in both lists of factors. The common factors are 1, 3, 9, 27, and 81.

$\boxed{\text{Common factors of 81 and 243 are } 1, 3, 9, 27, 81.}$ The HCF is the Highest Common Factor. From 1, 3, 9, 27, 81, the largest is 81. We can also find HCF by multiplying common prime factors. The prime factors of 81 are $3 \times 3 \times 3 \times 3$. The prime factors of 243 are $3 \times 3 \times 3 \times 3 \times 3$. The common prime factors are four 3s. $HCF = 3 \times 3 \times 3 \times 3$ $= 81$ $\boxed{\text{HCF}(81, 243) = 81}$

Answer

(a) Common factors of 50 and 60 are 1, 2, 5, 10. HCF(50, 60) = 10. (b) Common factors of 140 and 275 are 1, 5. HCF(140, 275) = 5. (c) Common factor of 77 and 725 is 1. HCF(77, 725) = 1. (d) Common factors of 370 and 592 are 1, 2, 37, 74. HCF(370, 592) = 74. (e) Common factors of 81 and 243 are 1, 3, 9, 27, 81. HCF(81, 243) = 81.