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**Exercise 3.4****Ques 1: ****Find the common factors of: ****(a) 20 and 28 ****(b) 15 and 25 ****(c) 35 and 50 ****(d) 56 and 120 ****Ans: **(a) Factors of 20 = 1, 2, 4, 5, 10, 20

Factors of 28 = 1, 2, 4, 7, 14, 28

Common factors = 1, 2, 4

(b) Factors of 15 = 1, 3, 5, 15

Factors of 25 = 1, 5, 25

Common factors = 1, 5

(c) Factors of 35 = 1, 5, 7, 35

Factors of 50 = 1, 2, 5, 10, 25, 50

Common factors = 1, 5

(d) Factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56

Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 60, 120

Common factors = 1, 2, 4, 8**Ques 2: ****Find the common factors of: ****(a) 4, 8 and 12 ****(b) 5, 15 and 25 ****Ans: **(a) Factors of 4 = 1, 2, 4

Factors of 8 = 1, 2, 4, 8

Factors of 12 = 1, 2, 3, 4, 6, 12

Common factors of 4, 8 and 12 = 1, 2, 4

(b) Factors of 5 = 1, 5

Factors of 15 = 1, 3, 5, 15

Factors of 25 = 1, 5, 25

Common factors of 5, 15 and 25 = 1, 5**Ques 3: ****Find the first three common multiples of: ****(a) 6 and 8 ****(b) 12 and 18 ****Ans: **(a) Multiple of 6 = 6, 12, 18, 24, 30, 36, 42, 28, 54, 60, 72, …………

Multiple of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, …………………….

Common multiples of 6 and 8 = 24, 48, 72

(b) Multiple of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ………

Multiple of 18 = 18, 36, 54, 72, 90, 108, ………………………………

Common multiples of 12 and 18 = 36, 72, 108**Ques 4: Write all the numbers less than 100 which are common multiples of 3 and 4. ****Ans: **Multiple of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99

Multiple of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100

Common multiples of 3 and 4 = 12, 24, 36, 48, 60, 72, 84, 96**Ques 5: Which of the following numbers are co-prime: ****(a) 18 and 35 ****(b) 15 and 37 ****(c) 30 and 415 ****(d) 17 and 68 ****(e) 216 and 215 ****(f) 81 and 16 ****Ans: **(a) Factors of 18 = 1, 2, 3, 6, 9, 18

Factors of 35 = 1, 5, 7, 35

Common factor = 1

Since, both have only one common factor, i.e., 1, therefore, they are co-prime numbers.

(b) Factors of 15 = 1, 3, 5, 15

Factors of 37 = 1, 37

Common factor = 1

Since, both have only one common factor, i.e., 1, therefore, they are co-prime numbers.

(c) Factors of 30 = 1, 2, 3, 5, 6, 15, 30

Factors of 415 = 1, 5, …….., 83, 415

Common factor = 1, 5

Since, both have more than one common factor, therefore, they are not co-prime numbers.

(d) Factors of 17 = 1, 17

Factors of 68 = 1, 2, 4, 17, 34, 86

Common factor = 1, 17

Since, both have more than one common factor, therefore, they are not co-prime numbers.

(e) Factors of 216 = 1, 2, 3, 4, 6, 8, 36, 72, 108, 216

Factors of 215 = 1, 5, 43, 215

Common factor = 1

Since, both have only one common factor, i.e., 1, therefore, they are co-prime numbers.

(f) Factors of 81 = 1, 3, 9, 27, 81

Factors of 16 = 1, 2, 4, 8, 16

Common factor = 1

Since, both have only one common factor, i.e., 1, therefore, they are co-prime numbers.**Ques 6: A number is divisible by both 5 and 12. By which other number will that number be always divisible? ****Ans:** 5 x 12 = 60. The number must be divisible by 60.**Ques 7: A number is divisible by 12. By what other numbers will that number be divisible? ****Ans:** Factors of 12 are 1, 2, 3, 4, 6 and 12.

Therefore, the number also be divisible by 1, 2, 3 4 and 6.**Exercise 3.5****Ques 1: Which of the following statements are true: ****(a) If a number is divisible by 3, it must be divisible by 9. ****(b) If a number is divisible by 9, it must be divisible by 3. ****(c) If a number is divisible by 18, it must be divisible by both 3 and 6. ****(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90. ****(e) If two numbers are co-primes, at least one of them must be prime. ****(f) All numbers which are divisible by 4 must also by divisible by 8. ****(g) All numbers which are divisible by 8 must also by divisible by 4. ****(h) If a number is exactly divides two numbers separately, it must exactly divide their sum. ****(i) If a number is exactly divides the sum of two numbers, it must exactly divide the two numbers separately. ****Ans:** Statements (b), (c), (d), (g) and (h) are true.**Ques 2: ****Here are two different factor trees for 60. Write the missing numbers. ****(a) **

**(b) **

**Answer 2: **

**(a) **

**(b) **

**Ques 3: Which factors are not included in the prime factorization of a composite number? ****Ans: **1 is the factor which is not included in the prime factorization of a composite number.**Ques 4: Write the greatest 4-digit number and express it in terms of its prime factors. ****Ans: **The greatest 4-digit number = 9999

The prime factors of 9999 are 3 × 3 × 11 × 101.**Ques 5: Write the smallest 5-digit number and express it in terms of its prime factors. ****Ans: **The smallest five digit number is 10000.

The prime factors of 10000 are 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5.**Ques 6: Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any, between, two consecutive prime numbers. ****Ans:** Prime factors of 1729 are 7 × 13 × 19.

The difference of two consecutive prime factors is 6.**Ques 7: The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples. ****Ans: **Among the three consecutive numbers, there must be one even number and one multiple of 3. Thus, the product must be multiple of 6.

Example: (i) 2 x 3 x 4 = 24

(ii) 4 x 5 x 6 = 120**Ques 8: The sum of two consecutive odd numbers is always divisible by 4. Verify this statement with the help of some examples. ****Ans: **3 + 5 = 8 and 8 is divisible by 4.

5 + 7 = 12 and 12 is divisible by 4.

7 + 9 = 16 and 16 is divisible by 4.

9 + 11 = 20 and 20 is divisible by 4.**Ques 9: In which of the following expressions, prime factorization has been done: ****(a) 24 = 2 x 3 x 4 ****(b) 56 = 7 x 2 x 2 x 2 ****(c) 70 = 2 x 5 x 7 ****(d) 54 = 2 x 3 x 9 ****Ans: **In expressions (b) and (c), prime factorization has been done.**Ques 10: Determine if 25110 is divisible by 45. ****[Hint: 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9.] ****Ans: **The prime factorization of 45 = 5 x 9

25110 is divisible by 5 as ‘0’ is at its unit place.

25110 is divisible by 9 as sum of digits is divisible by 9.

Therefore, the number must be divisible by 5 x 9 = 45**Ques 11: 18 is divisible by both 2 and 3. It is also divisible by 2 x 3 = 6. Similarly, a number is divisible by 4 and 6. Can we say that the number must be divisible by 4 x 6 = 24? If not, give an example to justify your answer. ****Ans:** No. Number 12 is divisible by both 6 and 4 but 12 is not divisible by 24.**Ques 12: I am the smallest number, having four different prime factors. Can you find me? ****Ans:** The smallest four prime numbers are 2, 3, 5 and 7.

Hence, the required number is 2 x 3 x 5 x 7 = 210

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