**Playing with Numbers**

**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.

(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 be divisible by 8.

(g) All numbers which are divisible by 8 must also be 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.

Answers:

(a) If a number is divisible by 3, it must be divisible by 9.

False.

6 is divisible by 3 but not divisible by 9.

(b) If a number is divisible by 9, it must be divisible by 3.

True.

(c) A number is divisible by 18 if it is divisible by both 3 and 6.

False

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

True.

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

False.

14 and 15 are co-primes, but both are composite numbers.

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

False.

20 is divisible by 4 but not divisible by 8.

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

True.

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

True.

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

False.

9 + 6 = 15. 15 is divisible by 5, but 9 and 6 are not divisible by 5.

2). Here are two different factor trees for 60. Write the missing numbers.

(a)

(b)

Answers:

a).

b).

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

1 and the number itself are not included in the prime factorization of a composite number.

4). Write the greatest 4-digit number and express it in terms of its prime factors.

Greatest 4-digit number = 9999

9999 = 9 X 1111

= 3 X 3 X 1111

= 3 X 3 X 11 X 101

Prime factors of the greatest 4-digit numbers are 3 X 3 X 11 X 101

5). Write the smallest 5-digit number and express it in the form of its prime factors.

smallest 5-digit number = 10000

10000 = 2X 5000

= 2 X 2 X 2500

= 2 X 2 X 2 X 1250

= 2 X 2 X 2 X 2 X 625

= 2 X 2 X 2 X 2 X 5 X 125

= 2 X 2 X 2 X 2 X 5 X 5 X 25

= 2 X 2 X 2 X 2 X 5 X 5 X 5 X 5

Prime factors of smallest 5-digit number 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.

1729 = 7 X 247

= 7 X 13 X 19

Prime factors of 1729 are 7, 13 and 19.

19 – 13 = 6 and 13- 7 = 6.

The difference between consecutive prime factors is 6.

7). The product of three consecutive numbers is always divisible by 6.

Verify this statement with the help of some examples.

The product of three consecutive numbers is always divisible by 6

Let take three consecutive numbers 3, 4 and 5.

3 X 4 X 5 = 60 it is divisible by 6.

Let us take three consecutive numbers 6, 7, 8

6 X 7 X 8 = 336 it is divisible by 6.

8). The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.

The sum of two consecutive odd numbers is divisible by 4.

Let us take two consecutive odd numbers 3 and 5.

3 + 8 = 8 s divisible by 4.

Let us take two consecutive odd numbers 7 and 9.

7 + 9 = 16 s divisible by 4.

9). In which of the following expressions, prime factorisation has been done?

(a) 24 = 2 × 3 × 4

(b) 56 = 7 × 2 × 2 × 2

(c) 70 = 2 × 5 × 7

(d) 54 = 2 × 3 × 9

Answer:

The prime factorization has been done in example (b) and (c).

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].

0 is in the unit’s place

Therefore, 25110 is divisible by 5.

2+5+1+1+0 = 9 Therefore, it is divisible by 9.

9 and 5 are co-prime numbers therefore 25110 is also divisible by (9 X 5 ) that is 45.

11). 18 is divisible by both 2 and 3. It is also divisible by 2 × 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 × 6 = 24? If not, give an example to justify your answer.

No

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

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

The four smallest consecutive prime numbers are 2 ,3, 5 and 7.

2 X 3 X 5 X 7 = 210.

The required number is 210.