**Playing with Numbers**

**Exercise 3.5**

**Ex 3.5 Question 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.**

**Solution:**

(a) False

(b) True

(c) False

(d) True

(e) False

(f) False

(g) True

(h) True

(i) False

**Ex 3.4 Question 2.**

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

**Solution:**

**(a)**

**(b) **

**Ex 3.4 Question 3.**

**Which factors are not included in the prime factorization of a composite number?**

**Solution:**

1 is the factor that is not included in the prime factorization of a composite number.

**Ex 3.4 Question 4.**

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

**Solution:**

The greatest 4-digit number = 9999

Hence, the prime factors of 9999 = 3 x 3 x 11 x 101.

**Ex 3.4 Question 5.**

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

**Solution:**

The smallest 5-digit number = 10000

Hence, the required prime factors: 10000 = 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5.

**Ex 3.4 Question 6.**

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

**Solution:**

Hence, the prime factors of 1729 = 7 x 13 x 19.

Here, 13 – 7 = 6 and 19 – 13 = 6

The difference between two consecutive prime factors is 6.

**Ex 3.4 Question 7.**

**The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.**

**Solution:**

Example 1:

Take three consecutive numbers 2, 3 and 4.

2 x 3 x 4 = 24

Therefore, the product 2 x 3 x 4 = 24 is divisible by 6.

Example 2:

Take three consecutive numbers 4 ,5 and 6.

4 x 5 x 6 = 120

Therefore, the product 4 x 5 x 6 = 120 is divisible by 6.

**Ex 3.4 Question 8.**

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

**Solution:**

Example 1:

5 + 3 = 8 and 8 is divisible by 4.

Example 2:

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

Example 3:

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

**Ex 3.4 Question 9.**

**In which of the following expressions, prime factorisation 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.**

**Solution:**

(a) 24 = 2 x 3 x 4

Here, 4 is not a prime number.

Hence, 24 = 2 x 3 x 4 is not a prime factorisation.

(b) 56 = 7 x 2 x 2 x 2

Here, all factors are prime numbers

Hence, 56 = 7 x 2 x 2 x 2 is a prime factorisation.

(c) 70 = 2 x 5 x 7

Here, all factors are prime numbers.

Hence, 70 = 2 x 5 x 7 is a prime factorisation.

(d) 54 = 2 x 3 x 9

Here, 9 is not a prime number.

Hence, 54 = 2 x 3 x 9 is not a prime factorisation.

**Ex 3.4 Question 10.**

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

**Solution:**

45 = 5 x 9

Here, 5 and 9 are co-prime numbers.

Test of divisibility by 5: a unit place of the given number 25110 is 0. So, it is divisible by 5.

Test of divisibility by 9:

Sum of the digits = 2 + 5 + l + l + 0 = 9 which is divisible by 9.

So, the given number is divisible by 5 and 9 both. Hence, the number 25110 is divisible by 45.

**Ex 3.4 Question 11.**

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

**Solution:**

The given two numbers are not co-prime. So, it is not necessary that a number divisible by both 4 and 6, must also be divisible by their product 4 x 6 = 24.

**Ex 3.4 Question 12.**

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

**Solution:**

The smallest 4 prime numbers = 2, 3, 5, and 7.

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