EXERCISE 3.5

**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) A number is divisible by 18 if it is divisible by both 3 and 6

d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.

d) If two numbers are co-prime, at least one of them must be prime.

e) All numbers which are divisible by 4 must also be divisible by 8.

f) All numbers which are divisible by 8 must also be divisible by 4.

g) The sum two consecutive odd numbers is divisible by 4.

h) If a number exactly divides two numbers separately, it must exactly divide their sum.

i) If a number exactly divides the sum of two numbers. it must exactly divide the two numbers separately

**Ans.** a) false b) true c) false d) true e) false

f) false g) true h) true i) true j) false

**2). Here are two different factor trees of 60. Write the missing number**

a)

b)

**3). Which factors are not included in the prime factorisation of a composite number?**

**Ans.** 1 and the number itself.

**4). Write the greatest four digit number and express it in term of its prime factors.**

**Ans.** The greatest four digit number is 9999. Its prime factor are = 3 x 3 x 11 x 101

**5). Write the smallest five digit number and express it in the form of its prime factor.**

**Ans.** The smallest five digit number = 10000 Its prime factors are = 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5

**6). Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.**

**Ans.** The prime factors of 1729 = 7 x 13 x 19. The difference between 2 consecutive

factors is 6

13 – 7 = 6

19 – 13 = 6

**7). The product of three consecutive numbers is always divisible by 6. Explain this statement with help of some examples.**

**Ans**. 2 x 3 x 4 = 24 which is divisible by 6. 7 x 8 x 9 = 504 which is divisible by 6.

**8). In which of the following expression, prime factor has been done:**

a) 24 = 2 x 3 x 4

**Ans**. Since 4 is composite, prime factorization has not been done.

b) 56 = 1 x 7 x 2 x 2 x 2

**Ans**. Since all the factors are prime, prime factorization has been done.

c) 70 = 2 x 5 x 7

**Ans**. Since all factors are prime, prime factorization has been done.

d) 54 = 2 x 3 x 9

**Ans**. Since 9 is a composite number, prime factorization has not been done.

**9). Write the prime factorization of 15470**

**Ans.** 15470 = 2 x 5 x 7 x 13 x 17

**10). Determine if 25110 is divisible by 45.**

**Ans.** 45 = 5 x 9

Factors of 5 = 1, 5

Factors of 9 = 1, 3, 9

Common factor = 1

∴5 and 9 are co- prime numbers.

The last digit of 25110 is 0, hence it is divisible by 5.

Sum of the digits 2 + 5 + 1 + 1 + 0 = 9

As 9 is divisible by 9 ∴ the number 25110 is divisible by 9.since the number is divisible by

both 5 and 9 ∴25110 is divisible by 45.

**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 4 and 6 but 12 is not divisible by 24.

**12). I am the smallest number, having four different prime factors. Can you find me?**

**Ans.** Since it is the smallest number it will be the product of 4 smallest prime numbers 2 x 3 x 5 x 7 = 210